1 Introduction
The subsequent inequalities are notable in the literature as Hermite–Hadamard’s inequality and Simpson’s inequality, respectively.
Theorem 1.1
Suppose that
\(f:I\subseteq \mathbb{R}\rightarrow \mathbb{R}\)
is a convex function defined on the interval
I
of real numbers and
\(a,b\in I\)
along with
\(a< b\). The following double inequality holds:
$$ f \biggl(\frac{a+b}{2} \biggr)\leq \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x\leq \frac{f(a)+f(b)}{2}. $$
Advertisement
Theorem 1.2
Assume that
\(f:[a,b]\rightarrow \mathbb{R}\)
is a four times continuously differentiable mapping on
\((a,b)\)
and
\(\Vert f^{(4)} \Vert _{\infty }=\mathrm{sup}_{x\in (a,b)} \vert f^{(4)}(x) \vert <\infty \). Then the following inequality holds:
$$ \biggl\vert \frac{1}{3} \biggl[ \frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \leq \frac{1}{2880} \bigl\Vert f^{(4)} \bigr\Vert _{\infty }(b-a)^{4}. $$
Hermite–Hadamard’s inequalities and Simpson’s inequalities have remained an area of great interest owing to their extensive applications in mathematics and other sciences. Many researchers generalized these inequalities. For recent results, for example, see [1‐8] and the references mentioned in these papers.
In 2013, Sarikaya et al. established the subsequent interesting Hermite–Hadamard’s inequalities by utilizing Riemann–Liouville fractional integrals.
Theorem 1.3
([9])
Advertisement
Let
\(f:[a,b]\rightarrow \mathbb{R}\)
be a positive function along with
\(0\leq a< b\), and let
\(f\in L^{1}[a,b]\). Suppose that
f
is a convex function on
\([a,b]\), then the following inequalities for fractional integrals hold:
where the symbols
\(J^{\mu }_{a^{+}} f\)
and
\(J^{\mu }_{b^{-}} f\)
denote respectively the left-sided and right-sided Riemann–Liouville fractional integrals of order
\(\mu >0\)
defined by
and
Here, \(\Gamma (\mu )\)
is the gamma function and its definition is
\(\Gamma (\mu )=\int _{0}^{\infty }e^{-t}t^{\mu -1}\,\mathrm{d}t\). It is to be noted that
\(J^{0}_{a^{+}}f(x)=J^{0}_{b^{-}}f(x)=f(x)\).
$$ f \biggl(\frac{a+b}{2} \biggr) \leq \frac{\Gamma (\mu +1)}{2(b-a)^{\mu }}\bigl[J^{\mu }_{a^{+}}f(b)+J^{\mu }_{b^{-}}f(a) \bigr]\leq \frac{f(a)+f(b)}{2}, $$
(1.1)
$$ J^{\mu }_{a^{+}}f(x) = \frac{1}{\Gamma (\mu )} \int ^{x}_{a}(x-t)^{\mu -1}f(t)\,\mathrm{d}t, \quad a< x, $$
$$ J^{\mu }_{b^{-}}f(x) = \frac{1}{\Gamma (\mu )} \int ^{b}_{x}(t-x)^{\mu -1}f(t)\,\mathrm{d}t, \quad x< b. $$
In the case of \(\mu =1\), the fractional integral recaptures the classical integral.
Because of the extensive application of Riemann–Liouville fractional integrals, some authors extended their studies to fractional Hermite–Hadamard’s inequalities via mappings of different classes. For example, refer to [10‐12] for convex mappings, to [13] for s-convex mappings, to [14] for \((s,m)\)-convex mappings, to [15] for s-Godunova–Levin mappings, to [16] for harmonically convex mappings, to [17] for preinvex mappings, to [18] for MT
m
-preinvex mappings, to [19] for h-convex mappings, to [20] for r-convex mappings, and see the references cited therein.
In 2012, Mubeen and Habibullah introduced the following class of fractional integrals.
Definition 1.1
([21])
Let \(f\in L^{1}[a,b]\), then the Riemann–Liouville k-fractional integrals \({}_{k}J^{\mu }_{a^{+}}f(x)\) and \({}_{k}J^{\mu }_{b^{-}}f(x)\) of order \(\mu >0\) are given as
and
respectively, where \(k>0\) and \(\Gamma _{k}(\mu )\) is the k-gamma function defined by \(\Gamma _{k}(\mu )=\int ^{\infty }_{0} t^{\mu -1} e^{-\frac{t^{k}}{k}}\,\mathrm{d}t\). Furthermore, \(\Gamma _{k}(\mu +k)=\mu \Gamma _{k}(\mu )\) and \({}_{k}J^{0}_{a^{+}}f(x)={}_{k}J^{0}_{b^{-}}f(x)=f(x)\).
$$ {}_{k}J^{\mu }_{a^{+}}f(x) = \frac{1}{k\Gamma _{k}(\mu )} \int ^{x}_{a}(x-t)^{\frac{\mu }{k}-1}f(t)\,\mathrm{d}t \quad (0\leq a< x< b) $$
$$ {} _{k}J^{\mu }_{b^{-}}f(x) = \frac{1}{k\Gamma _{k}(\mu )} \int ^{b}_{x}(t-x)^{\frac{\mu }{k}-1}f(t)\,\mathrm{d}t\quad (0\leq a< x< b), $$
The concept of Riemann–Liouville k-fractional integral is an important generalization of Riemann–Liouville fractional integrals. We would like to stress here that for \(k\neq 1\) the properties of Riemann–Liouville k-fractional integrals are very dissimilar to those of classical Riemann–Liouville fractional integrals. Due to this, the Riemann–Liouville k-fractional integrals have aroused many researchers’ interest. Properties and estimations for the integral inequality related to this operator can be sought out in [22‐27] and the references cited therein.
The main purpose of the current paper is to establish some new bounds on Hermite–Hadamard’s and Simpson’s inequalities for mappings whose absolute values of second derivatives are generalized \((m,h)\)-preinvex. To do this, the authors derive a general k-fractional integral identity along with multi parameters for twice differentiable mappings. By using this integral identity, the authors derive some new inequalities of Simpson and Hermite–Hadamard type for these mappings.
To end this section, we restate some special functions and definitions as follows.
Let us consider the following special functions:
(1)
The beta function:
$$ \begin{aligned} \beta (x,y)=\frac{\Gamma (x)\Gamma (y)}{\Gamma (x+y)}= \int ^{1}_{0}t^{x-1}(1-t)^{y-1}\,\mathrm{d}t, \quad x,y>0; \end{aligned} $$
(2)
The incomplete beta function:
$$ \begin{aligned} \beta (a;x,y)= \int ^{a}_{0}t^{x-1}(1-t)^{y-1}\,\mathrm{d}t, \quad 0< a< 1,x,y>0; \end{aligned} $$
(3)
The hypergeometric function:
$$ {}_{2}F_{1}(a,b;c;z)= \frac{1}{\beta (b,c-b)} \int ^{1}_{0} t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,\mathrm{d}t,\quad c>b>0,\vert z \vert < 1. $$
Definition 1.2
([28])
A function \(f : [0,\infty )\rightarrow \mathbb{R}\) is named s-convex in the second sense along with \(s\in (0,1]\) if
holds for all \(x,y\in [0,\infty )\) and \(\alpha ,\beta \geq 0\) along with \(\alpha +\beta =1\).
$$ \begin{aligned} f(\alpha x +\beta y)\leq \alpha ^{s} f(x)+ \beta ^{s} f(y) \end{aligned} $$
Definition 1.3
([29])
A function \(f :A \subseteq \mathbb{R}\rightarrow \mathbb{R}\) is called s-Godunova–Levin function of the second kind along with \(s\in [0,1]\) if
holds for all \(x,y\in A\) and \(t\in (0,1)\).
$$ \begin{aligned} f \bigl(tx+(1-t)y \bigr)\leq \frac{f(x)}{t^{s}}+ \frac{f(y)}{(1-t)^{s}} \end{aligned} $$
Definition 1.4
([30])
A function \(f:A\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is named \(tgs\)-convex on A if f is non-negative and
holds for all \(x, y \in A \) and \(t\in (0,1)\).
$$ f \bigl(tx+ (1-t)y \bigr)\leq t(1-t)\bigl[f(x)+f(y) \bigr] $$
Definition 1.5
([31])
A function \(f: A\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is called MT-convex if f is non-negative and
holds for all \(x,y\in A\) and \(t\in (0, 1)\).
$$ f \bigl(tx+(1-t)y \bigr)\leq \frac{\sqrt{t}}{2\sqrt{1-t}}f(x)+\frac{\sqrt{1-t}}{2\sqrt{t}}f(y) $$
Definition 1.6
([32])
A set \(A\subseteq \mathbb{R}^{n}\) is called m-invex with respect to the mapping \(\eta :A \times A\times (0, 1]\rightarrow \mathbb{R}^{n}\) for some fixed \(m\in (0, 1]\) if \(mx+\lambda \eta (y, x, m) \in A\) holds for all \(x, y \in A\) and \(\lambda \in [0, 1]\).
Definition 1.7
([33])
Let \(A \subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A \times A \times (0,1]\rightarrow \mathbb{R}\), and let \(h_{1}, h_{2} :[0,1]\rightarrow \mathbb{R}_{0}\). A function \(f:A\rightarrow \mathbb{R}\) is said to be generalized \((m, h_{1}, h_{2})\)-preinvex if
is valid for all \(x, y \in A\) and \(t \in [0, 1]\). If inequality (1.2) reverses, then f is said to be generalized \((m, h_{1}, h_{2})\)-preincave on A.
$$ f \bigl(m x+ t\eta (y, x, m) \bigr)\leq mh_{1}(t) f(x)+ h_{2}(t) f(y) $$
(1.2)
Clearly, if we take \(h_{1}(t)=h(1-t)\), \(h_{2}(t)=h(t)\) in Definition 1.7, then f becomes generalized \((m,h)\)-preinvex functions as follows.
Definition 1.8
Let \(A\subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\), and let \(h :[0,1]\rightarrow \mathbb{R}_{0}\). A function \(f:A\rightarrow \mathbb{R}\) is called generalized \((m,h)\)-preinvex if
is valid for all \(x, y \in A\) and \(t \in [0,1]\).
$$ f \bigl(mx+t\eta (y,x,m) \bigr)\leq h(t)f(y)+mh(1-t)f(x) $$
(1.3)
Remark 1.1
Let us discuss some special cases of Definition 1.8 as follows:
(i)
choosing \(h(t)=1\), we obtain the definition of generalized \((m,P)\)-preinvex functions;
(ii)
choosing \(h(t)=t^{s}\) for \(s\in (0,1]\), we obtain the definition of generalized \((m,s)\)-Breckner-preinvex functions;
(iii)
choosing \(h(t)=t^{-s}\) for \(s\in (0,1)\), we obtain the definition of generalized \((m,s)\)-Godunova–Levin–Dragomir-preinvex functions;
(iv)
choosing \(h(t)=t(1-t)\), we obtain the definition of generalized \((m, tgs)\)-preinvex functions;
(v)
choosing \(h(t)=\frac{\sqrt{t}}{2\sqrt{1-t}}\), we obtain the definition of generalized m-MT-preinvex functions.
It is worth mentioning here that, as far as we know, all the special cases considered above are new in the literature.
2 Main results
In order to derive our main results, we need the subsequent identity.
Lemma 2.1
Let
\(A\subseteq \mathbb{R}\)
be an open
m-invex subset with respect to
\(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\setminus \{0\}\)
for some fixed
\(m \in (0,1]\), and let
\(a, b \in A\), \(a< b\)
with
\(\eta (b,a,m)>0\). Assume that
\(f:A\rightarrow \mathbb{R}\)
is a twice differentiable function on
A
such that
\(f''\)
is integrable on
\([ma,ma+\eta (b,a,m) ]\). Then the following identity for Riemann–Liouville
k-fractional integrals along with
\(x \in [a, b]\), \(\lambda \in [0, 1]\), \(\mu > 0\), and
\(k>0\)
exists:
where
and
\(\Gamma _{k}\)
is the
k-gamma function.
$$\begin{aligned} &I_{f, \eta }(\mu ,k;x,\lambda ,m,a,b) \\ &\quad =\frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \int ^{1}_{0}t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr] f'' \bigl(ma+t \eta (x,a,m) \bigr)\,\mathrm{d}t \\ &\quad \quad {} +\frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \int ^{1}_{0}t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr] f'' \bigl(mb+t \eta (x,b,m) \bigr)\,\mathrm{d}t, \end{aligned}$$
(2.1)
$$\begin{aligned} &I_{f, \eta }(\mu ,k;x,\lambda ,m,a,b) \\ &\quad:=\frac{1-\lambda }{\eta (b,a,m)} \bigl[\eta ^{\frac{\mu }{k}}(x,a,m)f \bigl(ma+\eta (x,a,m) \bigr) \\ &\quad\quad {} +(-1)^{\frac{\mu }{k}}\eta ^{\frac{\mu }{k}}(x,b,m)f \bigl(mb+\eta (x,b,m) \bigr) \bigr] \\ &\quad\quad {} +\frac{\lambda }{\eta (b,a,m)} \bigl[\eta ^{\frac{\mu }{k}}(x,a,m)f(ma) +(-1)^{\frac{\mu }{k}}\eta ^{\frac{\mu }{k}}(x,b,m)f(mb) \bigr] \\ &\quad\quad {} +\frac{\frac{1}{\frac{\mu }{k}+1}-\lambda }{\eta (b,a,m)} \bigl[(-1)^{\frac{\mu }{k}+1}\eta ^{\frac{\mu }{k}+1}(x,b,m)f' \bigl(mb+\eta (x,b,m) \bigr) \\ &\quad\quad {} -\eta ^{\frac{\mu }{k}+1}(x,a,m)f' \bigl(ma+\eta (x,a,m) \bigr) \bigr] \\ &\quad \quad -\frac{\Gamma _{k}(\mu +k)}{\eta (b,a,m)} \bigl[{}_{k}J^{\mu }_{(ma+\eta (x,a,m))^{-}} f(ma)+{}_{k}J^{\mu }_{(mb+\eta (x,b,m))^{+}}f(mb) \bigr] \end{aligned}$$
Proof
By integration by parts and replacing the variable, we can state
Similarly, we get
Multiplying both sides of (2.2) and (2.3) by \(\frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)}\) and \(\frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)}\), respectively, and adding the resulting identities together, we obtain the desired result. □
$$\begin{aligned} &\int ^{1}_{0}t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr]f'' \bigl(ma+t \eta (x,a,m) \bigr)\,\mathrm{d}t \\ &\quad =t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr] \frac{f' (ma+t\eta (x,a,m) )}{\eta (x,a,m)} \bigg| _{0}^{1} \\ &\quad \quad {} -\frac{\frac{\mu }{k}+1}{\eta (x,a,m)} \int ^{1}_{0} \bigl(\lambda -t^{\frac{\mu }{k}} \bigr)f' \bigl(ma+t\eta (x,a,m) \bigr)\,\mathrm{d}t \\ &\quad = \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -1 \biggr] \frac{f' (ma+\eta (x,a,m) )}{\eta (x,a,m)} \\ &\quad \quad {} -\frac{\frac{\mu }{k}+1}{\eta (x,a,m)} \biggl[ \bigl(\lambda -t^{\frac{\mu }{k}} \bigr) \frac{f (ma+t\eta (x,a,m) )}{\eta (x,a,m)} \bigg| _{0}^{1} \\ &\quad \quad {} + \int ^{1}_{0}\frac{\mu }{k}t^{\frac{\mu }{k}-1} \frac{f (ma+t\eta (x,a,m) )}{\eta (x,a,m)}\,\mathrm{d}t \biggr] \\ &\quad = \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -1 \biggr] \frac{f' (ma+\eta (x,a,m) )}{\eta (x,a,m)} \\ &\quad \quad {} +\frac{\frac{\mu }{k}+1}{\eta ^{2}(x,a,m)} \biggl[(1-\lambda )f \bigl(ma+\eta (x,a,m) \bigr)+ \lambda f(ma) \\ &\quad \quad {} -\frac{\frac{\mu }{k}}{\eta ^{\frac{\mu }{k}}(x,a,m)}\int ^{ma+\eta (x,a,m)}_{ma}(u-ma)^{\frac{\mu }{k}-1}f(u)\,\mathrm{d}u \biggr] \\ &\quad = \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -1 \biggr]\frac{f' (ma+\eta (x,a,m) )}{\eta (x,a,m)} \\ &\quad \quad {} +\frac{\frac{\mu }{k}+1}{\eta ^{2}(x,a,m)} \biggl[(1-\lambda )f \bigl(ma+\eta (x,a,m) \bigr)+ \lambda f(ma) \\ &\quad \quad {} -\frac{\Gamma _{k}(\mu +k)}{\eta ^{\frac{\mu }{k}}(x,a,m)}{}_{k}J^{\mu }_{(ma+\eta (x,a,m))^{-}}f(ma)\biggr]. \end{aligned}$$
(2.2)
$$\begin{aligned} &\int ^{1}_{0}t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr]f'' \bigl(mb+t \eta (x,b,m) \bigr)\,\mathrm{d}t \\ &\quad = \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -1 \biggr] \frac{f' (mb+\eta (x,b,m) )}{\eta (x,b,m)} \\ &\quad \quad {} +\frac{\frac{\mu }{k}+1}{\eta ^{2}(x,b,m)} \biggl[(1-\lambda )f \bigl(mb+\eta (x,b,m) \bigr)+ \lambda f(mb) \\ &\quad \quad {} -\frac{\Gamma _{k}(\mu +k)}{\eta ^{\frac{\mu }{k}}(x,b,m)}{}_{k}J^{\mu }_{(ma+\eta (x,b,m))^{-}}f(mb)\biggr]. \end{aligned}$$
(2.3)
Remark 2.1
In Lemma 2.1, if we put \(k=1\) and \(\eta (b,a,m)=b-ma\) along with \(m=1\), then we get the following identity:
which is proved by İşcan in [34]. Further, if we put \(\mu =1, \lambda =\frac{1}{2}\), and \(x=a\) or \(x=b\), then the above identity recaptures Lemma 1 in [35].
$$\begin{aligned} &(1-\lambda ) \biggl[\frac{(x-a)^{\mu }+(b-x)^{\mu }}{b-a} \biggr]f(x)+ \lambda \biggl[\frac{(x-a)^{\mu }f(a)+(b-x)^{\mu }f(b)}{b-a} \biggr] \\ &\quad \quad {} + \biggl(\frac{1}{\mu +1}-\lambda \biggr) \biggl[\frac{(b-x)^{\mu +1}-(x-a)^{\mu +1}}{b-a} \biggr]f'(x)-\frac{\Gamma (\mu +1)}{b-a} \bigl[J^{\mu }_{x^{-}}f(a)+J^{\mu }_{x^{+}}f(b) \bigr] \\ &\quad =\frac{(x-a)^{\mu +2}}{(\mu +1)(b-a)} \int ^{1}_{0}t \bigl[ (\mu +1 )\lambda -t^{\mu } \bigr]f'' \bigl(tx+(1-t)a \bigr)\,\mathrm{d}t \\ &\quad \quad {} +\frac{(b-x)^{\mu +2}}{(\mu +1)(b-a)} \int ^{1}_{0}t \bigl[ (\mu +1 )\lambda -t^{\mu } \bigr]f'' \bigl(tx+(1-t)b \bigr)\,\mathrm{d}t, \end{aligned}$$
Using Lemma 2.1, we now state the following theorem.
Theorem 2.1
Let
\(A\subseteq \mathbb{R}\)
be an open
m-invex subset with respect to
\(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\setminus \{0\}\)
for some fixed
\(m \in (0,1]\), and let
\(a, b \in A\), \(a< b\)
with
\(\eta (b,a,m)>0\). Assume that
\(f:A\rightarrow \mathbb{R}\)
is a twice differentiable function on
A
such that
\(f''\)
is integrable on
\([ma,ma+\eta (b,a,m) ]\). If
\(\vert f'' \vert ^{q}\)
for
\(q\geq 1\)
is a generalized
\((m,h)\)-preinvex function with respect to
η
and
\(h:[0,1]\rightarrow \mathbb{R}_{0}\), then the following inequality for
k-fractional integrals with
\(x \in [a,b]\), \(\lambda \in [0, 1]\), \(\mu > 0\), \(k>0\)
exists:
where
and
$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}C_{1}(k,\mu ,\lambda ;h) \\ &\quad \quad {} +m \bigl\vert f''(a) \bigr\vert ^{q}C_{2}(k,\mu ,\lambda ;h) \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}C_{1}(k, \mu ,\lambda ;h) \\ &\quad \quad {} +m \bigl\vert f''(b) \bigr\vert ^{q}C_{2}(k,\mu ,\lambda ;h) \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$
(2.4)
$$\begin{aligned} &C_{0}(k,\mu ,\lambda )= \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \,\mathrm{d}t \\ &\hphantom{C_{0}(k,\mu ,\lambda )}=\textstyle\begin{cases} \frac{\frac{\mu }{k}[(\frac{\mu }{k}+1)\lambda ]^{1+\frac{2k}{\mu }}}{\frac{\mu }{k}+2}-\frac{(\frac{\mu }{k}+1)\lambda }{2}+\frac{1}{\frac{\mu }{k}+2},&0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{(\frac{\mu }{k}+1)\lambda }{2}-\frac{1}{\frac{\mu }{k}+2},&\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \end{aligned}$$
(2.5)
$$\begin{aligned} & C_{1}(k,\mu ,\lambda ;h)= \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert h(t)\,\mathrm{d}t,\quad 0\leq \lambda \leq 1, \end{aligned}$$
(2.6)
$$ C_{2}(k,\mu ,\lambda ;h)= \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert h(1-t)\,\mathrm{d}t,\quad 0\leq \lambda \leq 1. $$
(2.7)
Proof
Applying Lemma 2.1 and the power mean inequality, we have
Since \(\vert f'' \vert ^{q}\) is generalized \((m,h)\)-preinvex on \([ma,ma+\eta (b,a,m) ]\), we get
and
where \(C_{0}(k,\mu ,\lambda )\), \(C_{1}(k,\mu ,\lambda ;h)\), and \(C_{2}(k,\mu ,\lambda ;h)\) are defined by (2.5)–(2.7), respectively. Hence, if we use (2.9) and (2.10) in (2.8), we can get the desired result. This completes the proof. □
$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda,m,a,b) \bigr\vert \\ &\quad\leq \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert \,\mathrm{d}t \\ &\quad\quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert \,\mathrm{d}t \\ &\quad\leq \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl( \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \,\mathrm{d}t \biggr)^{1-\frac{1}{q}} \\ &\quad\quad {} \times \biggl(\int ^{1}_{0}t \biggl\vert \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr)^{\frac{1}{q}} \\ &\quad\quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl( \int ^{1}_{0}t \biggl\vert \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \,\mathrm{d}t \biggr)^{1-\frac{1}{q}} \\ &\quad\quad {} \times \biggl(\int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr)^{\frac{1}{q}} \\ &\quad =C_{0}^{1-\frac{1}{q}}(k,\mu,\lambda ) \\ &\quad \quad {} \times \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl( \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr)^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl( \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda \\ &\quad \quad {} -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr)^{\frac{1}{q}} \biggr\} . \end{aligned}$$
(2.8)
$$\begin{aligned} &\int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \\ &\quad \leq \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl[h(t) \bigl\vert f''(x) \bigr\vert ^{q}+mh(1-t) \bigl\vert f''(a) \bigr\vert ^{q} \bigr]\,\mathrm{d}t \\ &\quad = \bigl\vert f''(x) \bigr\vert ^{q}C_{1}(k,\mu ,\lambda ;h)+m \bigl\vert f''(a) \bigr\vert ^{q}C_{2}(k, \mu ,\lambda ;h) \end{aligned}$$
(2.9)
$$\begin{aligned} & \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \\ &\quad \leq \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl[h(t) \bigl\vert f''(x) \bigr\vert ^{q}+mh(1-t) \bigl\vert f''(b) \bigr\vert ^{q} \bigr]\,\mathrm{d}t \\ &\quad = \bigl\vert f''(x) \bigr\vert ^{q}C_{1}(k,\mu ,\lambda ;h)+m \bigl\vert f''(b) \bigr\vert ^{q}C_{2}(k, \mu ,\lambda ;h), \end{aligned}$$
(2.10)
Let us point out some special cases of Theorem 2.1.
Corollary 2.1
In Theorem 2.1, if
\(\vert f'' \vert ^{q}\)
for
\(q\geq 1\)
is generalized
\((m,s)\)-Breckner-preinvex functions, then, for
\(s\in (0,1]\)
and
\(m\in (0,1]\), we have
where we use the fact that
and
\(C_{0}(k,\mu ,\lambda )\)
is defined by (2.5).
$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad\leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \\ &\quad\quad {} \times \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}T_{1}(k, \mu ,\lambda ;s)+m \bigl\vert f''(a) \bigr\vert ^{q}T_{2}(k,\mu ,\lambda ;s) \bigr]^{\frac{1}{q}} \\ &\quad\quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}T_{1}(k, \mu ,\lambda ;s)+m \bigl\vert f''(b) \bigr\vert ^{q}T_{2}(k,\mu ,\lambda ;s) \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$
$$\begin{aligned}& T_{1}(k,\mu ,\lambda ;s)= \textstyle\begin{cases} \frac{\frac{2\mu }{k} [ (\frac{\mu }{k}+1 )\lambda ]^{1+\frac{k(s+2)}{\mu }}}{(s+2)(\frac{\mu }{k}+s+2)}-\frac{(\frac{\mu }{k}+1)\lambda }{s+2}+\frac{1}{\frac{\mu }{k}+s+2}, &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{(\frac{\mu }{k}+1)\lambda }{s+2}-\frac{1}{\frac{\mu }{k}+s+2}, &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \\& T_{2}(k,\mu ,\lambda ;s)= \textstyle\begin{cases} \left [ \textstyle\begin{array}{@{}c@{}} \beta (\frac{\mu }{k}+2,s+1 )- (\frac{\mu }{k}+1 )\lambda \beta (2,s+1 )\\ {}+2 (\frac{\mu }{k}+1 )\lambda \beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};2,s+1 )\\ {}-2\beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{\mu }{k}+2,s+1 ) \end{array}\displaystyle \right ] , &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ (\frac{\mu }{k}+1 )\lambda \beta (2,s+1)-\beta (\frac{\mu }{k}+2,s+1 ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \end{aligned}$$
Corollary 2.2
In Theorem 2.1, if the mapping
\(\eta (b,a,m)=b-ma\)
along with
\(m=1\), taking
\(x=\frac{a+b}{2}\), then for
\(s\in (0,1]\), we have the following inequality for
s-convex functions:
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\lambda ,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[ \frac{f(a)+f(b)}{2} \biggr] \\ &\quad \quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1}(k,\mu ,\lambda ;s)+ \bigl\vert f''(a) \bigr\vert ^{q}T_{2}(k, \mu ,\lambda ;s) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1}(k,\mu , \lambda ;s)+ \bigl\vert f''(b) \bigr\vert ^{q}T_{2}(k,\mu ,\lambda ;s) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Remark 2.2
In Corollary 2.2,
(i)
taking \(\lambda =\frac{1}{2}\), we obtain
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k,\mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k, \mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(ii)
taking \(\lambda =\frac{1}{3}\), we obtain
Specially, if we put \(k=\mu =s=1\), then we obtain
It is noted that the result of the first inequality in (2.11) is proved by İşcan in [34], which is better than the result presented by Sarikaya et al. in [36, Theorem 6].
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{3},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr] \\ &\quad \quad {}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu ,\frac{1}{3} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k,\mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k, \mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{162} \biggl\{ \biggl[\frac{59}{96} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{37}{96} \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{59}{96} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \frac{37}{96} \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} \\ &\quad \leq \frac{(b-a)^{2}}{162} \biggl\{ \biggl(\frac{59\vert f''(a) \vert ^{q}+133\vert f''(b) \vert ^{q}}{192} \biggr)^{\frac{1}{q}} \\ &\quad \quad {} + \biggl(\frac{59\vert f''(b) \vert ^{q}+133\vert f''(a) \vert ^{q}}{192} \biggr)^{\frac{1}{q}} \biggr\} . \end{aligned}$$
(2.11)
Remark 2.3
Corollary 2.3
In Theorem 2.1, if
\(\vert f'' \vert ^{q}\)
for
\(q\geq 1\)
is generalized
\((m,s)\)-Godunova–Levin–Dragomir-preinvex functions, then, for
\(s\in (0,1)\)
and
\(m\in (0,1]\), we have
where we use the fact that
and
\(C_{0}(k,\mu ,\lambda )\)
is defined by (2.5).
$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \\ &\quad \quad {} \times \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}U_{1}(k, \mu ,\lambda ;s)+m \bigl\vert f''(a) \bigr\vert ^{q}U_{2}(k,\mu ,\lambda ;s) \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}U_{1}(k, \mu ,\lambda ;s)+m \bigl\vert f''(b) \bigr\vert ^{q}U_{2}(k,\mu ,\lambda ;s) \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$
$$\begin{aligned}& U_{1}(k,\mu ,\lambda ;s)= \textstyle\begin{cases} \frac{\frac{2\mu }{k} [ (\frac{\mu }{k}+1 )\lambda ]^{1+\frac{k(2-s)}{\mu }}}{(2-s)(\frac{\mu }{k}-s+2)}-\frac{(\frac{\mu }{k}+1)\lambda }{2-s}+\frac{1}{\frac{\mu }{k}-s+2}, &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{(\frac{\mu }{k}+1)\lambda }{2-s}-\frac{1}{\frac{\mu }{k}-s+2}, &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \\& U_{2}(k,\mu ,\lambda ;s)= \textstyle\begin{cases} \left [ \textstyle\begin{array}{@{}c@{}} \beta (\frac{\mu }{k}+2,1-s )- (\frac{\mu }{k}+1 )\lambda \beta (2,1-s )\\ {}+2 (\frac{\mu }{k}+1 )\lambda \beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};2,1-s ) \\ {}-2\beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{\mu }{k}+2,1-s ) \end{array}\displaystyle \right ] , &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ (\frac{\mu }{k}+1 )\lambda \beta (2,1-s)-\beta (\frac{\mu }{k}+2,1-s ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \end{aligned}$$
Corollary 2.4
In Theorem 2.1, if the mapping
\(\eta (b,a,m)=b-ma\)
together with
\(m=1\), taking
\(x=\frac{a+b}{2}\), for
\(s\in (0,1)\), we have the following inequality for
s-Godunova–Levin functions:
$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] \\ &\quad \quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1}(k,\mu ,\lambda ;s)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2}(k, \mu ,\lambda ;s) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1}(k,\mu , \lambda ;s)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2}(k,\mu ,\lambda ;s) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Remark 2.4
In Corollary 2.4,
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
Specially, if we put \(k=1=\mu \), then we obtain a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{3},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{3} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k,\mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k, \mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},\frac{1}{3},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{8}{81} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2}\biggr) \biggr\vert ^{q}U_{1} \biggl(1,1, \frac{1}{3};s \biggr) \\ &\quad \quad {}+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{3};s \biggr)\biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(1,1,\frac{1}{3};s \biggr) + \bigl\vert f''(b)\bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
Specially, if we put \(k=1=\mu \), then we obtain an averaged midpoint-trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k,\mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k, \mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(1,1, \frac{1}{2};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(1,1,\frac{1}{2};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(c)
if \(\lambda =0\), then we obtain
Specially, if we put \(k=1=\mu \), then we obtain a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},0,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert f \biggl(\frac{a+b}{2} \biggr)-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{1}{\frac{\mu }{k}+2} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{\frac{\mu }{k}-s+2}+ \bigl\vert f''(a) \bigr\vert ^{q}\beta \biggl(\frac{\mu }{k}+2,1-s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{\frac{\mu }{k}-s+2}+ \bigl\vert f''(b) \bigr\vert ^{q}\beta \biggl(\frac{\mu }{k}+2,1-s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},0,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert f \biggl(\frac{a+b}{2} \biggr)-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{3} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{3-s}+ \bigl\vert f''(a) \bigr\vert ^{q}\beta (3,1-s ) \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{3-s}+ \bigl\vert f''(b) \bigr\vert ^{q}\beta (3,1-s ) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(d)
if \(\lambda =1\), then we obtain
Specially, if we put \(k=1=\mu \), then we obtain a trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},1,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}+3 )}{2 (\frac{\mu }{k}+2 )} \biggr]^{1-\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}-s+3 )\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s) (\frac{\mu }{k}-s+2 )} \\ &\quad \quad {} + \bigl\vert f''(a) \bigr\vert ^{q} \biggl( \biggl(\frac{\mu }{k}+1 \biggr)\beta (2,1-s)-\beta \biggl(\frac{\mu }{k}+2,1-s\biggr) \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}-s+3 )\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s) (\frac{\mu }{k}-s+2 )} \\ &\quad \quad {} + \bigl\vert f''(b) \bigr\vert ^{q} \biggl( \biggl(\frac{\mu }{k}+1 \biggr)\beta (2,1-s)-\beta \biggl(\frac{\mu }{k}+2,1-s \biggr) \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},1,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{2}{3} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[\frac{(4-s)\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s)(3-s)} + \bigl\vert f''(a) \bigr\vert ^{q} \bigl(2\beta (2,1-s)-\beta (3,1-s) \bigr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{(4-s)\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s)(3-s)} + \bigl\vert f''(b) \bigr\vert ^{q} \bigl(2\beta (2,1-s)-\beta (3,1-s) \bigr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Corollary 2.5
In Theorem 2.1, if
\(\vert f'' \vert ^{q}\)
for
\(q\geq 1\)
is generalized
\((m, tgs)\)-preinvex functions, then, for
\(m\in (0,1]\), we have
where we use the fact that
and
\(C_{0}(k,\mu ,\lambda )\)
is defined by (2.5).
$$ \begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda )\theta ^{\frac{1}{q}}(k, \mu ,\lambda ) \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(a) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(b) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned} $$
$$ \theta (k,\mu ,\lambda )= \textstyle\begin{cases} \frac{\frac{2\mu }{k} [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{3k}{\mu }+1}}{3(\frac{\mu }{k}+3)}-\frac{\frac{\mu }{k} [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{4k}{\mu }+1}}{2(\frac{\mu }{k}+4)}-\frac{(\frac{\mu }{k}+1)\lambda }{12}+\frac{1}{(\frac{\mu }{k}+3)(\frac{\mu }{k}+4)}, &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{(\frac{\mu }{k}+1)\lambda }{12}-\frac{1}{(\frac{\mu }{k}+3)(\frac{\mu }{k}+4)}, &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1\end{cases} $$
Corollary 2.6
In Theorem 2.1, if the mapping
\(\eta (b,a,m)=b-ma\)
along with
\(m=1\), taking
\(x=\frac{a+b}{2}\), we get the following inequality for
\(tgs\)-convex functions:
$$ \begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] \\ &\quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda )\theta ^{\frac{1}{q}}(k,\mu ,\lambda ) \\ &\quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
Remark 2.5
In Corollary 2.6,
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
Specially, if we put \(k=1=\mu \), then we obtain a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{3} \biggr)\theta ^{\frac{1}{q}} \biggl(k,\mu ,\frac{1}{3} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{162}{ \biggl(\frac{23}{160} \biggr)}^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$ \begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr)\theta ^{\frac{1}{q}} \biggl(k,\mu ,\frac{1}{2} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{96}{ \biggl(\frac{1}{5} \biggr)}^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(c)
if \(\lambda =0\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:
$$ \begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{1}{\frac{\mu }{k}+2} \biggr)^{1-\frac{1}{q}} \biggl[\frac{1}{(\frac{\mu }{k}+3)(\frac{\mu }{k}+4)} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{48} \biggl(\frac{3}{20} \biggr)^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(d)
if \(\lambda =1\), then we obtain
Specially, if we put \(k=1=\mu \), then we obtain a trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}+3 )}{2 (\frac{\mu }{k}+2 )} \biggr]^{1-\frac{1}{q}} \biggl[\frac{\frac{\mu }{k} (\frac{\mu ^{2}}{k^{2}}+\frac{8\mu }{k}+19 )}{12 (\frac{\mu }{k}+3 ) (\frac{\mu }{k}+4 )} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{24} \biggl(\frac{7}{40} \biggr)^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Corollary 2.7
In Theorem 2.1, if
\(\vert f'' \vert ^{q}\)
for
\(q\geq 1\)
is generalized
m-MT-preinvex functions, then, for
\(m\in (0,1]\), we have
where we use the fact that
and
\(C_{0}(k,\mu ,\lambda )\)
is defined by (2.5).
$$ \begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}\Phi _{1}(k,\mu ,\lambda )+m \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2}(k,\mu ,\lambda ) \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}\Phi _{1}(k,\mu ,\lambda )+m \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2}(k,\mu ,\lambda ) \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned} $$
$$\begin{aligned}& \Phi _{1}(k,\mu ,\lambda ) \\& \quad = \textstyle\begin{cases} \left [ \textstyle\begin{array}{@{}c@{}} \frac{1}{2}\beta (\frac{\mu }{k}+\frac{5}{2},\frac{1}{2} )-\frac{3\lambda \pi (\frac{\mu }{k}+1)}{16}+ (\frac{\mu }{k}+1 )\lambda \beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{5}{2},\frac{1}{2} )\\ {}-\beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{\mu }{k}+\frac{5}{2},\frac{1}{2} )\end{array}\displaystyle \right ] , &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{3\lambda \pi (\frac{\mu }{k}+1)}{16}-\frac{1}{2}\beta (\frac{\mu }{k}+\frac{5}{2},\frac{1}{2} ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \\& \Phi _{2}(k,\mu ,\lambda ) \\& \quad = \textstyle\begin{cases} \left [ \textstyle\begin{array}{@{}c@{}} \frac{1}{2}\beta (\frac{\mu }{k}+\frac{3}{2},\frac{3}{2} )-\frac{\lambda \pi (\frac{\mu }{k}+1)}{16} + (\frac{\mu }{k}+1 )\lambda \beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{3}{2},\frac{3}{2} )\\ {}-\beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{\mu }{k}+\frac{3}{2},\frac{3}{2} ) \end{array}\displaystyle \right ] , &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{\lambda \pi (\frac{\mu }{k}+1)}{16}-\frac{1}{2}\beta (\frac{\mu }{k}+\frac{3}{2},\frac{3}{2} ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \end{aligned}$$
Corollary 2.8
In Theorem 2.1, if the mapping
\(\eta (b,a,m)=b-ma\)
together with
\(m=1\), taking
\(x=\frac{a+b}{2}\), we get the following inequality for
MT-convex functions:
$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] \\ &\quad \quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1}(k,\mu ,\lambda )+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2}(k,\mu ,\lambda ) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1}(k,\mu ,\lambda )+ \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2}(k,\mu ,\lambda ) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Remark 2.6
In Corollary 2.8,
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:
where
and
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{0}^{1-\frac{1}{q}} \biggl(k,\mu ,\frac{1}{3} \biggr) \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{3} \biggr) \\ &\quad \quad {}+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{3} \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$ \begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} { \biggl(\frac{8}{81} \biggr)}^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(1,1,\frac{1}{3} \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2} \biggl(1,1,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(1,1,\frac{1}{3} \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2} \biggl(1,1,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \biggr\} , \end{aligned} $$
$$ \Phi _{1} \biggl(1,1,\frac{1}{3} \biggr)=\frac{71\sqrt{2}}{648}+\frac{1}{8}\arcsin {\frac{\sqrt{3}}{3}}- \frac{\pi }{32} $$
$$ \Phi _{2} \biggl(1,1,\frac{1}{3} \biggr)=\frac{19\sqrt{2}}{648}+\frac{1}{12}\arcsin {\frac{1}{3}}+ \frac{1}{16}\arctan {2\sqrt{2}}-\frac{\pi }{32}; $$
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr) \biggl\{ \biggl[ \biggl\vert f''\biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{2} \biggr) \\ &\quad \quad {}+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{2} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{2} \biggr)+ \bigl\vert f''(b)\bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{2} \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{96} \biggl(\frac{3\pi }{16} \biggr)^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(c)
if \(\lambda =0\), then we obtain
Specially, if we put \(k=1=\mu \), then we obtain a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{1}{\frac{\mu }{k}+2} \biggr)^{1-\frac{1}{q}} \biggl(\frac{1}{2} \biggr)^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{5}{2},\frac{1}{2} \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{5}{2},\frac{1}{2} \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{48} \biggl(\frac{3}{2} \biggr)^{\frac{1}{q}} \\ &\quad\quad {} \times \biggl\{ \biggl[\frac{5\pi }{16} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{\pi }{16} \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[\frac{5\pi }{16} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{\pi }{16} \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(d)
if \(\lambda =1\), then we obtain
Specially, if we put \(k=1=\mu \), then we obtain a trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{\frac{\mu }{k}(\frac{\mu }{k}+3)}{2(\frac{\mu }{k}+2)} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q} \biggl(\frac{3(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2} \beta \biggl(\frac{\mu }{k}+\frac{5}{2},\frac{1}{2} \biggr) \biggr) \\ &\quad \quad {} + \bigl\vert f''(a) \bigr\vert ^{q} \biggl(\frac{(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q} \biggl( \frac{3(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2}\beta \biggl(\frac{\mu }{k}+ \frac{5}{2},\frac{1}{2} \biggr) \biggr) \\ &\quad \quad {} + \bigl\vert f''(b) \bigr\vert ^{q} \biggl(\frac{(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{2}{3} \biggr)^{1-\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[\frac{7\pi }{32} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{3\pi }{32} \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\qquad {} + \biggl[\frac{7\pi }{32} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{3\pi }{32} \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Now, we get ready to state the second theorem as follows.
Theorem 2.2
Let
\(A\subseteq \mathbb{R}\)
be an open
m-invex subset with respect to
\(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\setminus \{0\}\)
for some fixed
\(m \in (0,1]\), and let
\(a, b \in A\), \(a< b\)
with
\(\eta (b,a,m)>0\). Assume that
\(f:A\rightarrow \mathbb{R}\)
is a twice differentiable function on
A
such that
\(f''\)
is integrable on
\([ma,ma+\eta (b,a,m) ]\). If
\(\vert f'' \vert ^{q}\)
for
\(q>1\)
is a generalized
\((m,h)\)-preinvex function with respect to
η
and
\(h:[0,1]\rightarrow \mathbb{R}_{0}\), then the following inequality for
k-fractional integrals with
\(x \in [a,b]\), \(\lambda \in [0, 1]\), \(\mu > 0\), \(k>0\)
exists:
where
\(p=\frac{q}{q-1}\)
and
$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \\ &\quad \quad {} \times \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[ \bigl\vert f''(x) \bigr\vert ^{q} \int ^{1}_{0}h(t)\,\mathrm{d}t+m \bigl\vert f''(a) \bigr\vert ^{q} \int ^{1}_{0}h(1-t)\,\mathrm{d}t \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \\ &\quad \quad {}\times \biggl[ \bigl\vert f''(x) \bigr\vert ^{q} \int ^{1}_{0}h(t)\,\mathrm{d}t+m \bigl\vert f''(b) \bigr\vert ^{q} \int ^{1}_{0}h(1-t)\,\mathrm{d}t \biggr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$
(2.12)
$$\begin{aligned} &C_{3}(k,\mu ,\lambda ,p) \\ &\quad = \textstyle\begin{cases} \frac{1}{p(\frac{\mu }{k}+1)+1},& \lambda =0, \\ \left [ \textstyle\begin{array}{@{}c@{}} \frac{k [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k+kp(\frac{\mu }{k}+1)}{\mu }}}{\mu }\beta (\frac{k(1+p)}{\mu },1+p )\\ {}+\frac{k [1- (\frac{\mu }{k}+1 )\lambda ]^{p+1}}{\mu (p+1)}{}_{2}F_{1} (1-\frac{k(1+p)}{\mu },1;p+2;1- (\frac{\mu }{k}+1 )\lambda ) \end{array}\displaystyle \right ] ,&0< \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{k [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k+kp(\frac{\mu }{k}+1)}{\mu }}}{\mu }\beta (\frac{1}{(\frac{\mu }{k}+1)\lambda };\frac{k(1+p)}{\mu },1+p ), & \frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1. \end{cases}\displaystyle \end{aligned}$$
Proof
Using Lemma 2.1 and Hölder’s inequality, we have
Since \(\vert f'' \vert ^{q}\) is generalized \((m,h)\)-preinvex on \([ma,ma+\eta (b,a,m)]\), we get
and
Hence, if we use (2.14)–(2.16) in (2.13), we can get the desired result. This completes the proof. □
$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert \,\mathrm{d}t \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert \,\mathrm{d}t \\ &\quad \leq \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[ \int ^{1}_{0}t^{p} \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert ^{p}\,\mathrm{d}t \biggr]^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl[ \int ^{1}_{0} \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[ \int ^{1}_{0}t^{p} \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert ^{p}\,\mathrm{d}t \biggr]^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl[ \int ^{1}_{0} \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr]^{\frac{1}{q}}. \end{aligned}$$
(2.13)
$$\begin{aligned} \int ^{1}_{0} \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t &\leq \int ^{1}_{0} \bigl[h(t) \bigl\vert f''(x) \bigr\vert ^{q}+mh(1-t) \bigl\vert f''(a) \bigr\vert ^{q} \bigr]\,\mathrm{d}t \\ &= \bigl\vert f''(x) \bigr\vert ^{q} \int ^{1}_{0}h(t)\,\mathrm{d}t+m \bigl\vert f''(a) \bigr\vert ^{q} \int ^{1}_{0}h(1-t)\,\mathrm{d}t, \end{aligned}$$
(2.14)
$$\begin{aligned} \int ^{1}_{0} \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t &\leq \int ^{1}_{0} \bigl[h(t) \bigl\vert f''(x) \bigr\vert ^{q}+mh(1-t) \bigl\vert f''(b) \bigr\vert ^{q} \bigr]\,\mathrm{d}t \\ &= \bigl\vert f''(x) \bigr\vert ^{q} \int ^{1}_{0}h(t)\,\mathrm{d}t+m \bigl\vert f''(b) \bigr\vert ^{q} \int ^{1}_{0}h(1-t)\,\mathrm{d}t, \end{aligned}$$
(2.15)
$$\begin{aligned}& C_{3}(k,\mu ,\lambda ,p) \\& \quad = \int ^{1}_{0}t^{p} \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert ^{p}\,\mathrm{d}t \\& \quad =\textstyle\begin{cases} \int ^{1}_{0}t^{(\frac{\mu }{k}+1)p}\,\mathrm{d}t,& \lambda =0,\\ \left [ \textstyle\begin{array}{cc} \int ^{[(\frac{\mu }{k}+1)\lambda ]^{\frac{k}{\mu }}}_{0}t^{p}[(\frac{\mu }{k}+1)\lambda -t^{\frac{\mu }{k}}]^{p}\,\mathrm{d}t \\ +\int ^{1}_{[(\frac{\mu }{k}+1)\lambda ]^{\frac{k}{\mu }}}t^{p}[t^{\frac{\mu }{k}}-(\frac{\mu }{k}+1)\lambda ]^{p}\,\mathrm{d}t \end{array}\displaystyle \right ] , &0< \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \int ^{1}_{0}t^{p}[(\frac{\mu }{k}+1)\lambda -t^{\frac{\mu }{k}}]^{p}\,\mathrm{d}t, &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1 \end{cases}\displaystyle \\& \quad = \textstyle\begin{cases} \frac{1}{p(\frac{\mu }{k}+1)+1},& \lambda =0, \\ \left [ \textstyle\begin{array}{@{}c@{}} \frac{k [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k+kp(\frac{\mu }{k}+1)}{\mu }}}{\mu }\beta (\frac{k(1+p)}{\mu },1+p )\\ {}+\frac{k [1- (\frac{\mu }{k}+1 )\lambda ]^{p+1}}{\mu (p+1)}{}_{2}F_{1} (1-\frac{k(1+p)}{\mu },1;p+2;1- (\frac{\mu }{k}+1 )\lambda ) \end{array}\displaystyle \right ] ,&0< \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{k [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k+kp(\frac{\mu }{k}+1)}{\mu }}}{\mu }\beta (\frac{1}{(\frac{\mu }{k}+1)\lambda };\frac{k(1+p)}{\mu },1+p ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1. \end{cases}\displaystyle \end{aligned}$$
(2.16)
Let us point out some special cases of Theorem 2.2.
Corollary 2.9
In Theorem 2.2, if we use the generalized
\((m,s)\)-Breckner-preinvexity of
\(\vert f'' \vert ^{q}\)
along with
\(q>1\)
and
\(p=\frac{q}{q-1}\), then, for
\(s\in (0,1]\)
and
\(m\in (0,1]\), we have the following inequality:
$$ \begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[\frac{\vert f''(x) \vert ^{q}+m\vert f''(a) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[ \frac{\vert f''(x) \vert ^{q}+m\vert f''(b) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
Corollary 2.10
In Theorem 2.2, if the mapping
\(\eta (b,a,m)=b-ma\)
together with
\(m=1\), choosing
\(x=\frac{a+b}{2}\), for
\(s\in (0,1]\), we have the following inequality for
s-convex functions:
$$ \begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\lambda ,1,a,b \biggr) \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \biggl\{ \biggl[\frac{\vert f''(x) \vert ^{q}+\vert f''(a) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} + \biggl[ \frac{\vert f''(x) \vert ^{q}+\vert f''(b) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
Remark 2.7
In Corollary 2.10,
(i)
if \(\lambda =\frac{1}{2}\), then we obtain
$$ \begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{3}^{\frac{1}{p}} \biggl(k,\mu , \frac{1}{2},p \biggr) \biggl\{ \biggl[\frac{\vert f''(x) \vert ^{q}+\vert f''(a) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} + \biggl[\frac{\vert f''(x) \vert ^{q}+\vert f''(b) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned} $$
Corollary 2.11
In Theorem 2.2, if we use the generalized
\((m,s)\)-Godunova–Levin–Dragomir-preinvexity of
\(\vert f'' \vert ^{q}\)
along with
\(q>1\)
and
\(p=\frac{q}{q-1}\), then, for
\(s\in (0,1)\)
and
\(m\in (0,1]\), we have the following inequality:
$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p)\frac{1}{{(1-s)}^{\frac{1}{q}}} \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(a) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(b) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Corollary 2.12
In Theorem 2.2, if the mapping
\(\eta (b,a,m)=b-ma\)
along with
\(m=1\), choosing
\(x=\frac{a+b}{2}\), for
\(s\in (0,1)\), we have the following inequality for
s-Godunova–Levin functions:
$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Remark 2.8
In Corollary 2.12,
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}} \biggl(k,\mu , \frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}} \biggl(1,1, \frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}} \biggl(k,\mu , \frac{1}{2},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}{\beta ^{\frac{1}{p}}(1+p,1+p)} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(c)
if \(\lambda =0\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}{ \biggl[\frac{1}{p (\frac{\mu }{k}+1 )+1} \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}{ \biggl(\frac{1}{2p+1} \biggr)}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(d)
if \(\lambda =1\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}}(k,\mu ,1,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}{ \biggl[2^{1+2p}\beta \biggl(\frac{1}{2};1+p,1+p \biggr) \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Corollary 2.13
In Theorem 2.2, if we use the generalized
\((m,tgs)\)-preinvexity of
\(\vert f'' \vert ^{q}\)
along with
\(q>1\)
and
\(p=\frac{q}{q-1}\), then, for
\(m\in (0,1]\), we have the following inequality:
where
\(C_{3}(k,\mu ,\lambda ,p)\)
is defined by (2.16).
$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p){ \biggl( \frac{1}{6} \biggr)}^{\frac{1}{q}} \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(a) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(b) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$
Corollary 2.14
In Theorem 2.2, if the mapping
\(\eta (b,a,m)=b-ma\)
together with
\(m=1\), choosing
\(x=\frac{a+b}{2}\), we get the following inequality for
\(tgs\)-convex functions:
$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k, \mu ,\lambda ,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Remark 2.9
In Corollary 2.14,
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(1,1, \frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$ \begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{2},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
$$ \begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}\beta ^{\frac{1}{p}}(1+p,1+p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned} $$
(c)
if \(\lambda =0\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}{ \biggl[\frac{1}{p (\frac{\mu }{k}+1 )+1} \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}{ \biggl(\frac{1}{2p+1} \biggr)}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(d)
if \(\lambda =1\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a trapezoid-type inequality:
$$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k, \mu ,1,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
$$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}{ \biggl[2^{1+2p}\beta \biggl(\frac{1}{2};1+p,1+p \biggr) \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
Corollary 2.15
In Theorem 2.2, if we use the generalized
m-MT-preinvexity of
\(\vert f'' \vert ^{q}\)
along with
\(q>1\)
and
\(p=\frac{q}{q-1}\), then, for
\(m\in (0,1]\), we have the following inequality:
where we use the fact that
$$ \begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \biggl( \frac{\pi }{4} \biggr)^{\frac{1}{q}} \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(a) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(b) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned} $$
$$ \begin{aligned} \int ^{1}_{0}\frac{\sqrt{t}}{2\sqrt{1-t}}\,\mathrm{d}t= \int ^{1}_{0}\frac{\sqrt{1-t}}{2\sqrt{t}}\,\mathrm{d}t= \frac{\pi }{4}. \end{aligned} $$
Corollary 2.16
In Theorem 2.2, if the mapping
\(\eta (b,a,m)=b-ma\)
together with
\(m=1\), choosing
\(x=\frac{a+b}{2}\), we get the following inequality for
MT-convex functions:
$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] \\ &\quad \quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Remark 2.10
In Corollary 2.16,
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr] \\ &\quad \quad {}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(1,1,\frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}}\biggr\} ; \end{aligned}$$
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr] \\ &\quad \quad {}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{2},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}\beta ^{\frac{1}{p}}(1+p,1+p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
(c)
if \(\lambda =0\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}{ \biggl[\frac{1}{p (\frac{\mu }{k}+1 )+1} \biggr]}^{\frac{1}{p}} \\ &\quad\quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
$$ \begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}{ \biggl(\frac{1}{2p+1} \biggr)}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned} $$
(d)
if \(\lambda =1\), then we obtain
Specially, if we put \(k=1=\mu \), then we derive a trapezoid-type inequality:
$$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k, \mu ,1,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}{ \biggl[2^{1+2p}\beta \biggl(\frac{1}{2};1+p,1+p \biggr) \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China under Grant No. 61374028.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.