1 Introduction
Nearly no mathematician can ignore the significant role of convex sets and convex functions in applied mathematics, especially in nonlinear programming and optimization theory. In addition, the elegance of convex sets and functions in shape and property makes studying this branch of mathematical analysis attractive. On the other hand, it should be observed that in new problems related to convexity, generalized notions for convex sets and functions are required to reach favorite and applicable results. In the past six decades, many determined attempts have gone in universalizing the concept of convexity. One of such generalizations is the concept of quasiconvex functions given in the definition below.
Unless otherwise stated, we shall assume throughout this paper that \(\mathfrak {D}\subseteq \mathbb {R}\) is an interval, and \(\mathfrak {D}^{\circ }\) represents the interior of \(\mathfrak {D}\).
Anzeige
Definition 1
([6])
A function \(\mathcal {K}:\mathfrak {D}\to \mathbb {R}\) is called quasiconvex on \(\mathfrak {D}\) if
for all \(\mathfrak {u},\mathfrak {v}\in \mathfrak {D}\) and \(\tau \in [0,1]\).
$$ \mathcal {K}\bigl(\tau \mathfrak {u}+(1-\tau )\mathfrak {v}\bigr)\leq \max \bigl\{ \mathcal {K}(\mathfrak {u}), \mathcal {K}(\mathfrak {v}) \bigr\} $$
For this class of quasiconvex functions, Ion established the following results.
Theorem 2
([11])
Anzeige
Let
\(\mathcal {K}:\mathfrak {D}\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\), \(\mathfrak {\alpha },\beta \in \mathfrak {D}^{\circ }\)
with
\(\mathfrak {\alpha }<\beta \). If
\(|\mathcal {K}'|\)
is quasiconvex on
\([\mathfrak {\alpha },\beta ]\), then
$$ \biggl\vert \frac{\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta )}{2}-\frac{1}{\beta -\mathfrak {\alpha }} \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r}) \,d\mathfrak {r}\biggr\vert \leq \frac{\beta -\mathfrak {\alpha }}{4}\max \bigl\{ \bigl\vert \mathcal {K}'(\mathfrak {\alpha }) \bigr\vert , \bigl\vert \mathcal {K}'(\beta ) \bigr\vert \bigr\} . $$
(1)
Theorem 3
([11])
Let
\(\mathcal {K}:\mathfrak {D}\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\), \(\mathfrak {\alpha },\beta \in \mathfrak {D}^{\circ }\)
with
\(\mathfrak {\alpha }<\beta \). If
\(|\mathcal {K}'|^{p/(p-1)}\)
is quasiconvex on
\([\mathfrak {\alpha },\beta ]\)
for
\(p>1\), then
$$ \begin{aligned}[b] & \biggl\vert \frac{\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta )}{2}- \frac{1}{\beta -\mathfrak {\alpha }} \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r}) \,d\mathfrak {r}\biggr\vert \\ &\quad \leq \frac{\beta -\mathfrak {\alpha }}{2(p+1)^{1/p}} \bigl[\max \bigl\{ \bigl\vert \mathcal {K}'(\mathfrak {\alpha }) \bigr\vert ^{p/(p-1)}, \bigl\vert \mathcal {K}'(\beta ) \bigr\vert ^{p/(p-1)} \bigr\} \bigr] ^{(p-1)/p}. \end{aligned} $$
(2)
Following the same line of thought, Alomari et al. proved the following.
Theorem 4
([1])
Let
\(\mathcal {K}:\mathfrak {D}\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\), \(\mathfrak {\alpha },\beta \in \mathfrak {D}^{\circ }\)
with
\(\mathfrak {\alpha }<\beta \). If
\(|\mathcal {K}'|^{q}\)
is quasiconvex on
\([\mathfrak {\alpha },\beta ]\)
for
\(q\geq 1\), then
$$ \begin{aligned}[b] & \biggl\vert \frac{\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta )}{2}- \frac{1}{\beta -\mathfrak {\alpha }} \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r}) \,d\mathfrak {r}\biggr\vert \leq \frac{\beta -\mathfrak {\alpha }}{4} \bigl[\max \bigl\{ \bigl\vert \mathcal {K}'(\mathfrak {\alpha }) \bigr\vert ^{q}, \bigl\vert \mathcal {K}'(\beta ) \bigr\vert ^{q} \bigr\} \bigr]^{1/q}. \end{aligned} $$
(3)
In another direction, Özdemir et al. obtained an estimate for the integral \(\int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}(\beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\) in the following theorem.
Theorem 5
([22])
Let
\(\mathcal {K}:[\mathfrak {\alpha },\beta ]\subset [0,\infty )\to \mathbb {R}\)
be continuous on
\([\mathfrak {\alpha },\beta ]\)
such that
\(\mathcal {K}\in L([\mathfrak {\alpha },\beta ])\)
with
\(\mathfrak {\alpha }<\beta \). If
\(\mathcal {K}\)
is quasiconvex on
\([\mathfrak {\alpha },\beta ]\), then, for some fixed
\(p,q>0\), we obtain
where
\(\mathcal{B}(\cdot ,\cdot )\)
is the Beta function defined for any
\(m, n>0\)
as thus:
$$ \begin{aligned}[b] &\int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \mathcal{B}(p+1,q+1)\max \bigl\{ \mathcal {K}( \mathfrak {\alpha }),\mathcal {K}(\beta )\bigr\} , \end{aligned} $$
(4)
$$ \mathcal{B}(m,n)= \int _{0}^{1}\mathfrak {u}^{m-1}(1-\mathfrak {u})^{n-1}\,d\mathfrak {u}. $$
Motivated by the above results, Liu proved the following results in same direction.
Theorem 6
([16])
Let
\(\mathcal {K}:[\mathfrak {\alpha },\beta ]\subset [0,\infty )\to \mathbb {R}\)
be continuous on
\([\mathfrak {\alpha },\beta ]\)
such that
\(\mathcal {K}\in L([\mathfrak {\alpha },\beta ])\)
with
\(\mathfrak {\alpha }<\beta \), and let
\(x>1\)
with
\(1/x+1/y=1\). If
\(|\mathcal {K}|^{y}\)
is quasiconvex on
\([\mathfrak {\alpha },\beta ]\), then, for some fixed
\(p,q>0\), we obtain
$$ \begin{aligned}[b] & \int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \bigl[\mathcal{B}(px+1,qx+1) \bigr]^{\frac{1}{x}} \bigl[\max \bigl\{ \bigl\vert \mathcal {K}(\mathfrak {\alpha }) \bigr\vert ^{y}, \bigl\vert \mathcal {K}(\beta ) \bigr\vert ^{y} \bigr\} \bigr]^{\frac{1}{y}}. \end{aligned} $$
(5)
Theorem 7
Let
\(\mathcal {K}:[\mathfrak {\alpha },\beta ]\subset [0,\infty )\to \mathbb {R}\)
be continuous on
\([\mathfrak {\alpha },\beta ]\)
such that
\(\mathcal {K}\in L([\mathfrak {\alpha },\beta ])\)
with
\(\mathfrak {\alpha }<\beta \), and let
\(s\geq 1\). If
\(|\mathcal {K}|^{s}\)
is quasiconvex on
\([\mathfrak {\alpha },\beta ]\), then, for some fixed
\(p,q>0\), we obtain
$$ \begin{aligned}[b] &\int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \mathcal{B}(p+1,q+1) \bigl[\max \bigl\{ \bigl\vert \mathcal {K}(\mathfrak {\alpha }) \bigr\vert ^{s}, \bigl\vert \mathcal {K}( \beta ) \bigr\vert ^{s} \bigr\} \bigr]^{\frac{1}{s}}. \end{aligned} $$
(6)
Recently, Gordji et al. introduced a new class of convexity—called η-quasiconvex functions.
Definition 8
([6])
A function \(\mathcal {K}:\mathfrak {D}\to \mathbb {R}\) is called η-quasiconvex on \(\mathfrak {D}\) with respect to \(\eta :\mathbb {R}\times \mathbb {R}\to \mathbb {R}\) if
for all \(\mathfrak {u},\mathfrak {v}\in \mathfrak {D}\) and \(\tau \in [0,1]\).
$$ \mathcal {K}\bigl(\tau \mathfrak {u}+(1-\tau )\mathfrak {v}\bigr)\leq \max \bigl\{ \mathcal {K}(\mathfrak {v}), \mathcal {K}(\mathfrak {v})+\eta \bigl(\mathcal {K}(\mathfrak {u}),\mathcal {K}(\mathfrak {v})\bigr) \bigr\} $$
A generalization of the above definition was recently proposed by Awan et al. as follows.
Definition 9
([2])
A function \(\mathcal {K}:\mathfrak {D}\to \mathbb {R}\) is called strongly η-quasiconvex on \(\mathfrak {D}\) with respect to \(\eta :\mathbb {R}\times \mathbb {R}\to \mathbb {R}\) and nonnegative modulus μ if
for all \(\mathfrak {u},\mathfrak {v}\in \mathfrak {D}\) and \(\tau \in [0,1]\).
$$ \mathcal {K}\bigl(\tau \mathfrak {u}+(1-\tau )\mathfrak {v}\bigr)\leq \max \bigl\{ \mathcal {K}(\mathfrak {v}), \mathcal {K}(\mathfrak {v})+\eta \bigl(\mathcal {K}(\mathfrak {u}),\mathcal {K}(\mathfrak {v})\bigr) \bigr\} - \mu \tau (1-\tau ) (\mathfrak {v}-\mathfrak {u})^{2} $$
Some papers concerning functions that are η-quasiconvex and strongly η-quasiconvex have been published. We refer the interested reader to [4, 5, 7, 8, 13, 17‐19, 21] and the references cited therein.
Inspired by the above results, it is our goal, in this present article, to:
1.
2 Main results
For the proof of our main results, the following lemmas will be useful.
Lemma 10
([23])
Let
\(\mathcal {K}:\mathfrak {D}\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\), \(\mathfrak {\alpha }, \beta \in \mathfrak {D}^{\circ }\)
with
\(\mathfrak {\alpha }\leq \beta \). If
\(\mathcal {K}''\in L([\mathfrak {\alpha },\beta ])\), then the following equality holds:
$$ \begin{aligned}[b] & \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \frac{1}{2}(\beta -\mathfrak {\alpha }) \bigl(\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta ) \bigr)+ \frac{5}{4}(\beta -\mathfrak {\alpha })^{2} \bigl(\mathcal {K}'(\beta )-\mathcal {K}'(\mathfrak {\alpha }) \bigr) \\ &\quad = \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \int _{0}^{1}(\tau +1)^{2} \bigl[ \mathcal {K}'' \bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr)+ \mathcal {K}'' \bigl(\tau \beta +(1-\tau )\mathfrak {\alpha }\bigr) \bigr]\,d\tau . \end{aligned} $$
(7)
Lemma 11
([22])
Let
\(\mathcal {K}:[\mathfrak {\alpha },\beta ]\subset [0,\infty )\to \mathbb {R}\)
be continuous on
\([\mathfrak {\alpha },\beta ]\)
such that
\(\mathcal {K}\in L([\mathfrak {\alpha },\beta ])\)
with
\(\mathfrak {\alpha }<\beta \). Then for some fixed
\(p,q>0\)
the following equality holds:
$$ \begin{aligned} \int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}=(\beta -\mathfrak {\alpha })^{p+q+1} \int _{0}^{1}(1-\tau )^{p}\tau ^{q}\mathcal {K}\bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr)\,d\tau . \end{aligned} $$
(8)
2.1 Perturbed trapezoid inequalities via η-quasiconvexity
We now frame and justify our results for the class of η-quasiconvex functions.
Theorem 12
Let
\(\mathcal {K}:\mathfrak {D}\subset [0,\infty )\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\)
such that
\(\mathcal {K}''\in L([\mathfrak {\alpha },\beta ])\), where
\(\mathfrak {\alpha },\beta \in \mathfrak {D}\)
with
\(\mathfrak {\alpha }<\beta \). If
\(|\mathcal {K}''|\)
is an
η-quasiconvex function on
\([\mathfrak {\alpha },\beta ]\), then
where
\(\mathbf{M}^{\beta }_{\mathfrak {\alpha }}(|\mathcal {K}''|,\eta )\)
and
\(\mathbf{N}^{\beta } _{\mathfrak {\alpha }}(|\mathcal {K}''|,\eta )\)
are defined by (10) and (11), respectively.
$$ \begin{aligned}[b] & \biggl\vert \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \frac{1}{2}(\beta -\mathfrak {\alpha }) \bigl(\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta ) \bigr)+ \frac{5}{4}(\beta -\mathfrak {\alpha })^{2} \bigl(\mathcal {K}'(\beta )-\mathcal {K}'(\mathfrak {\alpha }) \bigr) \biggr\vert \\ &\quad \leq \frac{7(\beta -\mathfrak {\alpha })^{3}}{12} \bigl(\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert , \eta \bigr)+\mathbf{N}^{\beta }_{\mathfrak {\alpha }}\bigl( \bigl\vert \mathcal {K}'' \bigr\vert ,\eta \bigr) \bigr), \end{aligned} $$
(9)
Proof
By using the definition of η-quasiconvexity of \(|\mathcal {K}''|\) on \([\mathfrak {\alpha },\beta ]\), one obtains the following inequalities:
and
for \(\tau \in [0, 1]\). Now, applying Lemma 10, and inequalities (10) and (11), we get
That completes the proof of Theorem 12. □
$$ \begin{aligned}[b] \bigl\vert \mathcal {K}'' \bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert &\leq \max \bigl\{ \bigl\vert \mathcal {K}''( \beta ) \bigr\vert , \bigl\vert \mathcal {K}''(\beta ) \bigr\vert +\eta \bigl( \bigl\vert \mathcal {K}''( \mathfrak {\alpha }) \bigr\vert , \bigl\vert \mathcal {K}''(\beta ) \bigr\vert \bigr) \bigr\} \\ &=:\mathbf{M}^{\beta }_{\mathfrak {\alpha }}\bigl( \bigl\vert \mathcal {K}'' \bigr\vert ,\eta \bigr) \end{aligned} $$
(10)
$$ \begin{aligned}[b] \bigl\vert \mathcal {K}'' \bigl(\tau \beta +(1-\tau )\mathfrak {\alpha }\bigr) \bigr\vert &\leq \max \bigl\{ \bigl\vert \mathcal {K}''( \mathfrak {\alpha }) \bigr\vert , \bigl\vert \mathcal {K}''(\mathfrak {\alpha }) \bigr\vert +\eta \bigl( \bigl\vert \mathcal {K}''( \beta ) \bigr\vert , \bigl\vert \mathcal {K}''(\mathfrak {\alpha }) \bigr\vert \bigr) \bigr\} \\ &=:\mathbf{N}^{\beta }_{\mathfrak {\alpha }}\bigl( \bigl\vert \mathcal {K}'' \bigr\vert ,\eta \bigr) \end{aligned} $$
(11)
$$\begin{aligned} & \biggl\vert \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \frac{1}{2}(\beta -\mathfrak {\alpha }) \bigl(\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta ) \bigr)+ \frac{5}{4}(\beta -\mathfrak {\alpha })^{2} \bigl(\mathcal {K}'(\beta )-\mathcal {K}'(\mathfrak {\alpha }) \bigr) \biggr\vert \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \int _{0}^{1}(\tau +1)^{2} \bigl\vert \mathcal {K}'' \bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr)+\mathcal {K}'' \bigl(\tau \beta +(1-\tau )\mathfrak {\alpha }\bigr) \bigr\vert \,d\tau \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \int _{0}^{1}(\tau +1)^{2} \bigl[ \bigl\vert \mathcal {K}'' \bigl(\tau \mathfrak {\alpha }+(1-\tau ) \beta \bigr) \bigr\vert + \bigl\vert \mathcal {K}'' \bigl(\tau \beta +(1-\tau )\mathfrak {\alpha }\bigr) \bigr\vert \bigr]\,d\tau \\ &\quad= \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \bigl(\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert , \eta \bigr)+\mathbf{N}^{\beta }_{\mathfrak {\alpha }}\bigl( \bigl\vert \mathcal {K}'' \bigr\vert ,\eta \bigr) \bigr) \int _{0}^{1}(\tau +1)^{2} \,d \tau \\ &\quad= \frac{7(\beta -\mathfrak {\alpha })^{3}}{12} \bigl(\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert , \eta \bigr)+\mathbf{N}^{\beta }_{\mathfrak {\alpha }}\bigl( \bigl\vert \mathcal {K}'' \bigr\vert ,\eta \bigr) \bigr). \end{aligned}$$
Theorem 13
Let
\(\mathcal {K}:\mathfrak {D}\subset [0,\infty )\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\)
such that
\(\mathcal {K}''\in L([\mathfrak {\alpha },\beta ])\), where
\(\mathfrak {\alpha },\beta \in \mathfrak {D}\)
with
\(\mathfrak {\alpha }<\beta \), and let
\(p>1\)
with
\(1/p+1/q=1\). If
\(|\mathcal {K}''|^{q}\)
is an
η-quasiconvex function on
\([\mathfrak {\alpha },\beta ]\), then
where
\(\mathbf{M}^{\beta }_{\mathfrak {\alpha }}(|\mathcal {K}''|^{q},\eta )\)
and
\(\mathbf{N}^{\beta } _{\mathfrak {\alpha }}(|\mathcal {K}''|^{q},\eta )\)
are defined by (12) and (13), respectively.
$$\begin{aligned} & \biggl\vert \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \frac{1}{2}(\beta -\mathfrak {\alpha }) \bigl(\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta ) \bigr)+ \frac{5}{4}(\beta -\mathfrak {\alpha })^{2} \bigl(\mathcal {K}'(\beta )-\mathcal {K}'(\mathfrak {\alpha }) \bigr) \biggr\vert \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \biggl(\frac{2^{2p+1}-1}{2p+1} \biggr) ^{\frac{1}{p}} \bigl[ \bigl(\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q}, \eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl( \mathbf{N}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$
Proof
The η-quasiconvexity of the function \(|\mathcal {K}''|^{q}\) implies that, for \(\tau \in [0, 1]\),
and
$$ \begin{aligned}[b] \bigl\vert \mathcal {K}'' \bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert ^{q} &\leq \max \bigl\{ \bigl\vert \mathcal {K}''(\beta ) \bigr\vert ^{q}, \bigl\vert \mathcal {K}''(\beta ) \bigr\vert ^{q}+\eta \bigl( \bigl\vert \mathcal {K}''( \mathfrak {\alpha }) \bigr\vert ^{q}, \bigl\vert \mathcal {K}''( \beta ) \bigr\vert ^{q} \bigr) \bigr\} \\ &=:\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr) \end{aligned} $$
(12)
$$ \begin{aligned}[b] \bigl\vert \mathcal {K}'' \bigl(\tau \beta +(1-\tau )\mathfrak {\alpha }\bigr) \bigr\vert ^{q} &\leq \max \bigl\{ \bigl\vert \mathcal {K}''(\mathfrak {\alpha }) \bigr\vert ^{q}, \bigl\vert \mathcal {K}''(\mathfrak {\alpha }) \bigr\vert ^{q}+\eta \bigl( \bigl\vert \mathcal {K}''( \beta ) \bigr\vert ^{q}, \bigl\vert \mathcal {K}''( \mathfrak {\alpha }) \bigr\vert ^{q} \bigr) \bigr\} \\ &=:\mathbf{N}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr). \end{aligned} $$
(13)
Also, it is easy to see that
Using Lemma 10, Hölder’s inequality, and thereafter Eqs. (12)–(14), one obtains
That establishes the desired inequality. □
$$ \int _{0}^{1}(\tau +1)^{2p}\,d \tau =\frac{2^{2p+1}-1}{2p+1}. $$
(14)
$$\begin{aligned} & \biggl\vert \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \frac{1}{2}(\beta -\mathfrak {\alpha }) \bigl(\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta ) \bigr)+ \frac{5}{4}(\beta -\mathfrak {\alpha })^{2} \bigl(\mathcal {K}'(\beta )-\mathcal {K}'(\mathfrak {\alpha }) \bigr) \biggr\vert \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \biggl[ \int _{0}^{1}(\tau +1)^{2} \bigl\vert \mathcal {K}'' \bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert \,d\tau \\ &\qquad {}+ \int _{0}^{1}(\tau +1)^{2} \bigl\vert \mathcal {K}'' \bigl(\tau \beta +(1-\tau )\mathfrak {\alpha }\bigr) \bigr\vert \,d\tau \biggr] \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \biggl[ \biggl( \int _{0}^{1}(\tau +1)^{2p} \,d \tau \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} \bigl\vert \mathcal {K}'' \bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert ^{q}\,d\tau \biggr)^{\frac{1}{q}} \\ & \qquad {} + \biggl( \int _{0}^{1}(\tau +1)^{2p}\,d \tau \biggr)^{ \frac{1}{p}} \biggl( \int _{0}^{1} \bigl\vert \mathcal {K}'' \bigl(\tau \beta +(1-\tau )\mathfrak {\alpha }\bigr) \bigr\vert ^{q}\,d\tau \biggr)^{\frac{1}{q}} \biggr] \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \biggl(\frac{2^{2p+1}-1}{2p+1} \biggr) ^{\frac{1}{p}} \bigl[ \bigl(\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q}, \eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl( \mathbf{N}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}} \bigr]. \end{aligned}$$
Theorem 14
Let
\(\mathcal {K}:\mathfrak {D}\subset [0,\infty )\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\)
such that
\(\mathcal {K}''\in L([\mathfrak {\alpha },\beta ])\), where
\(\mathfrak {\alpha },\beta \in \mathfrak {D}\)
with
\(\mathfrak {\alpha }<\beta \). If
\(|\mathcal {K}''|^{q}\)
is an
η-quasiconvex function on
\([\mathfrak {\alpha },\beta ]\)
for
\(q\geq 1\), then
where
\(\mathbf{M}^{\beta }_{\mathfrak {\alpha }}(|\mathcal {K}''|^{q},\eta )\)
and
\(\mathbf{N}^{\beta } _{\mathfrak {\alpha }}(|\mathcal {K}''|^{q},\eta )\)
are defined as in Theorem 13.
$$\begin{aligned} & \biggl\vert \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \frac{1}{2}(\beta -\mathfrak {\alpha }) \bigl(\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta ) \bigr)+ \frac{5}{4}(\beta -\mathfrak {\alpha })^{2} \bigl(\mathcal {K}'(\beta )-\mathcal {K}'(\mathfrak {\alpha }) \bigr) \biggr\vert \\ &\quad \leq \frac{7(\beta -\mathfrak {\alpha })^{3}}{12} \bigl[ \bigl(\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl(\mathbf{N} ^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$
Proof
Applying Lemma 10 and the power integral inequality, one gets
Hence, the intended result is obtained. □
$$\begin{aligned} & \biggl\vert \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \frac{1}{2}(\beta -\mathfrak {\alpha }) \bigl(\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta ) \bigr)+ \frac{5}{4}(\beta -\mathfrak {\alpha })^{2} \bigl(\mathcal {K}'(\beta )-\mathcal {K}'(\mathfrak {\alpha }) \bigr) \biggr\vert \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \biggl[ \int _{0}^{1}(\tau +1)^{2} \bigl\vert \mathcal {K}'' \bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert \,d\tau \\ &\qquad {}+ \int _{0}^{1}(\tau +1)^{2} \bigl\vert \mathcal {K}'' \bigl(\tau \beta +(1-\tau )\mathfrak {\alpha }\bigr) \bigr\vert \,d\tau \biggr] \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \biggl( \int _{0}^{1}(\tau +1)^{2}\,d \tau \biggr) ^{1-\frac{1}{q}} \biggl[ \biggl( \int _{0}^{1} (\tau +1)^{2} \mathbf{M} ^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr)\,d \tau \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1} (\tau +1)^{2} \mathbf{N}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr)\,d \tau \biggr) ^{\frac{1}{q}}\biggr] \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{3}}{4} \int _{0}^{1}(\tau +1)^{2}\,d \tau \bigl[ \bigl(\mathbf{M} ^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl(\mathbf{N}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr) \bigr) ^{\frac{1}{q}} \bigr] \\ &\quad \leq \frac{7(\beta -\mathfrak {\alpha })^{3}}{12} \bigl[ \bigl(\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl(\mathbf{N} ^{\beta }_{\mathfrak {\alpha }} \bigl( \bigl\vert \mathcal {K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}} \bigr]. \end{aligned}$$
2.2 More inequalities via strong η-quasiconvexity
We now present some new estimates for \(\int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}(\beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r}) \,d\mathfrak {r}\) by means of the strong η-quasiconvexity.
Theorem 15
Let
\(\mathcal {K}:[\mathfrak {\alpha },\beta ]\subset [0,\infty )\to \mathbb {R}\)
be continuous on
\([\mathfrak {\alpha },\beta ]\)
such that
\(\mathcal {K}\in L([\mathfrak {\alpha },\beta ])\)
with
\(\mathfrak {\alpha }<\beta \). If
\(\mathcal {K}\)
is strongly
η-quasiconvex with nonnegative modulus
μ
on
\([\mathfrak {\alpha },\beta ]\), then, for some fixed
\(p,q>0\), we obtain
$$ \begin{aligned}[b] &\int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \bigl[{\mathbf{M}}^{\beta }_{\mathfrak {\alpha }}( \mathcal {K},\eta )\mathcal{B}(p+1,q+1)-\mu (\beta -\mathfrak {\alpha })^{2} \mathcal{B}(p+2,q+2) \bigr]. \end{aligned} $$
(15)
Proof
The strong η-quasiconvexity of \(\mathcal {K}\) on \([\mathfrak {\alpha },\beta ]\) implies that, for \(\tau \in [\mathfrak {\alpha },\beta ]\), one obtains
for \(\tau \in [0,1]\). From Lemma 11 and (16), we get
which gives the intended inequality. □
$$ \begin{aligned}[b] \mathcal {K}\bigl(\tau \mathfrak {\alpha }+(1-\tau ) \beta \bigr) \leq {}&\max \bigl\{ \mathcal {K}(\beta ), \mathcal {K}(\beta )+ \eta \bigl(\mathcal {K}(\mathfrak {\alpha }),\mathcal {K}(\beta ) \bigr) \bigr\} \\ &{} -\mu \tau (1-\tau ) (\beta -\mathfrak {\alpha })^{2} \\ =:{}&\mathbf{M}^{\beta }_{\mathfrak {\alpha }}(\mathcal {K},\eta )- \mu \tau (1- \tau ) (\beta -\mathfrak {\alpha })^{2} \end{aligned} $$
(16)
$$\begin{aligned} &\int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad =(\beta -\mathfrak {\alpha })^{p+q+1} \int _{0}^{1}(1-\tau )^{p}\tau ^{q}\mathcal {K}\bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr)\,d\tau \\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \int _{0}^{1}(1-\tau )^{p}\tau ^{q} \bigl[{\mathbf{M}}^{\beta }_{\mathfrak {\alpha }}(\mathcal {K}, \eta )-\mu \tau (1-\tau ) (\beta -\mathfrak {\alpha })^{2} \bigr]\,d\tau \\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \biggl[{\mathbf{M}}^{\beta }_{\mathfrak {\alpha }}( \mathcal {K},\eta ) \int _{0}^{1}(1-\tau )^{p}\tau ^{q}\,d\tau -\mu (\beta -\mathfrak {\alpha })^{2} \int _{0}^{1}\tau ^{q+1}(1-\tau )^{p+1}\,d\tau \biggr] \\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \bigl[{\mathbf{M}}^{\beta }_{\mathfrak {\alpha }}( \mathcal {K},\eta ) \mathcal{B}(p+1,q+1)-\mu (\beta -\mathfrak {\alpha })^{2} \mathcal{B}(p+2,q+2) \bigr], \end{aligned}$$
Remark 16
Let \(\eta (x,y)=x-y\). If, in addition, we assume that \(\mathcal {K}\) in Theorem 15 is
1.
increasing, then (15) boils down to
$$ \begin{aligned}& \int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \bigl[\mathcal{B}(p+1,q+1)\mathcal {K}(\beta )- \mu (\beta -\mathfrak {\alpha })^{2}\mathcal{B}(p+2,q+2) \bigr]; \end{aligned} $$
2.
decreasing, then (15) becomes
$$ \begin{aligned}& \int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \bigl[\mathcal{B}(p+1,q+1)\mathcal {K}(\mathfrak {\alpha })- \mu (\beta -\mathfrak {\alpha })^{2}\mathcal{B}(p+2,q+2) \bigr]. \end{aligned} $$
Theorem 17
Let
\(\mathcal {K}:[\mathfrak {\alpha },\beta ]\subset [0,\infty )\to \mathbb {R}\)
be continuous on
\([\mathfrak {\alpha },\beta ]\)
such that
\(\mathcal {K}\in L([\mathfrak {\alpha },\beta ])\)
with
\(\mathfrak {\alpha }<\beta \), and let
\(x>1\)
with
\(1/x+1/y=1\). If
\(|\mathcal {K}|^{y}\)
is strongly
η-quasiconvex with nonnegative modulus
μ
on
\([\mathfrak {\alpha },\beta ]\), then, for some fixed
\(p,q>0\), we obtain
$$ \begin{aligned}[b] & \int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \bigl[\mathcal{B}(px+1,qx+1) \bigr]^{\frac{1}{x}} \biggl[{\mathbf{M}}^{\beta } _{\mathfrak {\alpha }} \bigl( \vert \mathcal {K}\vert ^{y},\eta \bigr)-\mu \frac{(\beta -\mathfrak {\alpha })^{2}}{6} \biggr] ^{\frac{1}{y}}. \end{aligned} $$
(17)
Proof
The strong η-quasiconvexity of the function \(|\mathcal {K}|^{y}\) on \([\mathfrak {\alpha },\beta ]\) implies that, for \(\tau \in [0, 1]\),
Applying (18) to Lemma 11, Hölder’s inequality and the definition of the Beta function given above, one obtains
That completes the proof. □
$$ \begin{aligned}[b] \bigl\vert \mathcal {K}\bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert ^{y} \leq {}& \max \bigl\{ \bigl\vert \mathcal {K}(\beta ) \bigr\vert ^{y}, \bigl\vert \mathcal {K}(\beta ) \bigr\vert ^{y}+\eta \bigl( \bigl\vert \mathcal {K}(\mathfrak {\alpha }) \bigr\vert ^{y}, \bigl\vert \mathcal {K}(\beta ) \bigr\vert ^{y} \bigr) \bigr\} \\ &{} -\mu \tau (1-\tau ) (\beta -\mathfrak {\alpha })^{2} \\ =:{}&\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \vert \mathcal {K}\vert ^{y},\eta \bigr)-\mu \tau (1-\tau ) (\beta -\mathfrak {\alpha })^{2}. \end{aligned} $$
(18)
$$\begin{aligned} &\int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \int _{0}^{1}(1-\tau )^{p}\tau ^{q} \bigl\vert \mathcal {K}\bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert \,d\tau \\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \biggl[ \int _{0}^{1}(1-\tau )^{px}\tau ^{qx}\,d\tau \biggr] ^{\frac{1}{x}} \biggl[ \int _{0}^{1} \bigl\vert \mathcal {K}\bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert ^{y}\,d\tau \biggr]^{\frac{1}{y}} \\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \biggl[ \int _{0}^{1}(1-\tau )^{px}\tau ^{qx}\,d\tau \biggr]^{\frac{1}{x}} \biggl[ \int _{0}^{1} \bigl(\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \vert \mathcal {K}\vert ^{y},\eta \bigr)- \mu \tau (1-\tau ) (\beta -\mathfrak {\alpha })^{2} \bigr)\,d\tau \biggr]^{\frac{1}{y}} \\ &\quad =(\beta -\mathfrak {\alpha })^{p+q+1} \bigl[\mathcal{B}(px+1,qx+1) \bigr]^{\frac{1}{x}} \biggl[{\mathbf{M}}^{\beta } _{\mathfrak {\alpha }} \bigl( \vert \mathcal {K}\vert ^{y},\eta \bigr)-\mu \frac{(\beta -\mathfrak {\alpha })^{2}}{6} \biggr] ^{\frac{1}{y}}. \end{aligned}$$
We end this section with our last result.
Theorem 18
Let
\(\mathcal {K}:[\mathfrak {\alpha },\beta ]\subset [0,\infty )\to \mathbb {R}\)
be continuous on
\([\mathfrak {\alpha },\beta ]\)
such that
\(\mathcal {K}\in L([\mathfrak {\alpha },\beta ])\)
with
\(\mathfrak {\alpha }<\beta \), and let
\(s\geq 1\). If
\(|\mathcal {K}|^{s}\)
is strongly
η-quasiconvex with nonnegative modulus
μ
on
\([\mathfrak {\alpha },\beta ]\), then, for some fixed
\(p,q>0\), we obtain
$$ \begin{aligned}[b] & \int _{\mathfrak {\alpha }}^{\beta } (\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \bigl[\mathcal{B}(p+1,q+1) \bigr]^{\frac{s-1}{s}} \\ &\qquad {}\times \bigl[{\mathcal{B}(p+1,q+1) \mathbf{M}}^{\beta }_{\mathfrak {\alpha }} \bigl( \vert \mathcal {K}\vert ^{s},\eta \bigr)-\mu (\beta -\mathfrak {\alpha })^{2} \mathcal{B}(p+2,q+2) \bigr]^{\frac{1}{s}}. \end{aligned} $$
(19)
Proof
We proceed as in the proof of Theorem 17, but this time we use the mean power inequality. We have
which gives the desired result. □
$$\begin{aligned} &\int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}( \beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}\\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \int _{0}^{1} \bigl[(1-\tau )^{p} \tau ^{q} \bigr]^{\frac{s-1}{s}} \bigl[(1-\tau )^{p} \tau ^{q} \bigr]^{\frac{1}{s}} \bigl\vert \mathcal {K}\bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert \,d\tau \\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \biggl[ \int _{0}^{1}(1-\tau )^{p}\tau ^{q}\,d\tau \biggr] ^{\frac{s-1}{s}} \biggl[ \int _{0}^{1}(1-\tau )^{p}\tau ^{q} \bigl\vert \mathcal {K}\bigl(\tau \mathfrak {\alpha }+(1-\tau )\beta \bigr) \bigr\vert ^{s}\,d\tau \biggr]^{\frac{1}{s}} \\ &\quad \leq (\beta -\mathfrak {\alpha })^{p+q+1} \biggl[ \int _{0}^{1}(1-\tau )^{p}\tau ^{q}\,d\tau \biggr]^{\frac{s-1}{s}} \\ &\qquad {}\times \biggl[ \int _{0}^{1}(1-\tau )^{p}\tau ^{q} \bigl(\mathbf{M}^{\beta }_{\mathfrak {\alpha }} \bigl( \vert \mathcal {K}\vert ^{s},\eta \bigr)-\mu \tau (1-\tau ) (\beta -\mathfrak {\alpha })^{2} \bigr)\,d\tau \biggr] ^{\frac{1}{s}} \\ &\quad =(\beta -\mathfrak {\alpha })^{p+q+1} \bigl[\mathcal{B}(p+1,q+1) \bigr]^{ \frac{s-1}{s}} \\ &\qquad {}\times \bigl[{\mathcal{B}(p+1,q+1)\mathbf{M}} ^{\beta }_{\mathfrak {\alpha }} \bigl( \vert \mathcal {K}\vert ^{s},\eta \bigr)-\mu (\beta -\mathfrak {\alpha })^{2} \mathcal{B}(p+2,q+2) \bigr]^{\frac{1}{s}}, \end{aligned}$$
3 Applications to trapezoidal formula
We start by recalling the following well-known result: let \(\mathcal {K}:[\mathfrak {\alpha },\beta ] \to \mathbb {R}\) be a function whose second derivative exists in \((\mathfrak {\alpha },\beta )\) and let P be a partition of the interval \([\mathfrak {\alpha },\beta ]\), that is, \(\mathbf{P}: \mathfrak {\alpha }=\mathfrak {r}_{1}<\mathfrak {r}_{2}<\cdots <\mathfrak {r}_{m-1}<\mathfrak {r}_{m}=\beta \), then
where the trapezoidal formula \(\mathbb {F}(\mathcal {K},\mathbf{P})\) is defined by
and the approximation error \(\mathbb {E}(\mathcal {K},\mathbf{P})\) of the integral \(\mathfrak {J}\) by the trapezoidal formula \(\mathbb {F}(\mathcal {K},\mathbf{P})\) and satisfies the inequality
with \(Q=\max_{\mathfrak {r}\in (\mathfrak {\alpha },\beta )}|\mathcal {K}''(\mathfrak {r})|<\infty \). In this section, we establish some perturbed new inequalities analogous to the estimate in (20) for twice differentiable functions.
$$ \mathfrak {J}:= \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}=\mathbb {F}( \mathcal {K},\mathbf{P})+\mathbb {E}(\mathcal {K}, \mathbf{P}), $$
$$ \mathbb {F}(\mathcal {K},\mathbf{P})=\sum_{k=0}^{m-1} \frac{\mathcal {K}(\mathfrak {r}_{k})+\mathcal {K}(\mathfrak {r}_{k+1})}{2}(\mathfrak {r}_{k+1}-\mathfrak {r}_{k}) $$
$$ \bigl\vert \mathbb {E}(\mathcal {K},\mathbf{P}) \bigr\vert \leq \frac{Q}{12}\sum_{k=0}^{m-1}( \mathfrak {r}_{k+1}-\mathfrak {r}_{k})^{3} $$
(20)
Proposition 20
Let
\(\mathcal {K}:\mathfrak {D}\subset [0,\infty )\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\)
such that
\(\mathcal {K}''\in L([\mathfrak {\alpha },\beta ])\), where
\(\mathfrak {\alpha },\beta \in \mathfrak {D}\)
with
\(\mathfrak {\alpha }<\beta \). If
\(|\mathcal {K}''|\)
is quasiconvex on
\([\mathfrak {\alpha },\beta ]\), then for every partition
P
of
\([\mathfrak {\alpha },\beta ]\)
where
\(\mathbb {G}(\mathcal {K},\mathbf{P})=\sum_{k=0}^{m-1}(\mathfrak {r}_{k+1}-\mathfrak {r}_{k})^{2}\frac{\mathcal {K}'(\mathfrak {r}_{k+1})-\mathcal {K}'(\mathfrak {r}_{k})}{4}\).
$$ \begin{aligned} & \bigl\vert \mathbb {E}(\mathcal {K}, \mathbf{P})+5\mathbb {G}(\mathcal {K},\mathbf{P}) \bigr\vert \leq \frac{7}{6} \sum_{k=0}^{m-1}(\mathfrak {r}_{k+1}- \mathfrak {r}_{k})^{3}\max \bigl\{ \bigl\vert \mathcal {K}''(\mathfrak {r}_{k}) \bigr\vert , \bigl\vert \mathcal {K}''(\mathfrak {r}_{k+1}) \bigr\vert \bigr\} , \end{aligned} $$
Proof
By using the function \(\eta (x,y)=x-y\) in (9) of Theorem 12, one gets
Now employing (21) for the subintervals \([\mathfrak {r}_{k},\mathfrak {r}_{k+1}]\) for \(k=\overline{0, m-1}\) of P, we obtain
Summing both sides of the above inequality over k from 0 to \(m-1\) amounts to
from which the desired inequality is obtained. □
$$ \begin{aligned}[b] & \biggl\vert \int _{\mathfrak {\alpha }}^{\beta }\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \frac{1}{2}(\beta -\mathfrak {\alpha }) \bigl(\mathcal {K}(\mathfrak {\alpha })+\mathcal {K}(\beta ) \bigr)+ \frac{5}{4}(\beta -\mathfrak {\alpha })^{2} \bigl(\mathcal {K}'(\beta )-\mathcal {K}'(\mathfrak {\alpha }) \bigr) \biggr\vert \\ &\quad \leq \frac{7(\beta -\mathfrak {\alpha })^{3}}{6} \max \bigl\{ \bigl\vert \mathcal {K}''(\mathfrak {\alpha }) \bigr\vert , \bigl\vert \mathcal {K}''(\beta ) \bigr\vert \bigr\} . \end{aligned} $$
(21)
$$ \begin{aligned} & \biggl\vert \int _{\mathfrak {r}_{k}}^{\mathfrak {r}_{k+1}}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \frac{1}{2}(\mathfrak {r}_{k+1}-\mathfrak {r}_{k}) \bigl(\mathcal {K}(\mathfrak {r}_{k})+\mathcal {K}(\mathfrak {r}_{k+1}) \bigr)+ \frac{5}{4}(\mathfrak {r}_{k+1}-\mathfrak {r}_{k})^{2} \bigl(\mathcal {K}'(\mathfrak {r}_{k+1})-\mathcal {K}'( \mathfrak {r}_{k}) \bigr) \biggr\vert \\ & \quad\leq \frac{7(\mathfrak {r}_{k+1}-\mathfrak {r}_{k})^{3}}{6} \max \bigl\{ \bigl\vert \mathcal {K}''(\mathfrak {r}_{k}) \bigr\vert , \bigl\vert \mathcal {K}''(\mathfrak {r}_{k+1}) \bigr\vert \bigr\} . \end{aligned} $$
$$ \begin{aligned} & \Biggl\vert \sum _{k=0}^{m-1} \int _{\mathfrak {r}_{k}}^{\mathfrak {r}_{k+1}}\mathcal {K}(\mathfrak {r})\,d\mathfrak {r}- \sum _{k=0}^{m-1}(\mathfrak {r}_{k+1}-\mathfrak {r}_{k}) \frac{\mathcal {K}(\mathfrak {r}_{k})+\mathcal {K}(\mathfrak {r}_{k+1})}{2} \\ &\qquad {}+5\sum_{k=0}^{m-1}( \mathfrak {r}_{k+1}-\mathfrak {r}_{k})^{2} \frac{\mathcal {K}'(\mathfrak {r}_{k+1})-\mathcal {K}'(\mathfrak {r}_{k})}{4}\Biggr\vert \\ &\quad\leq \frac{7}{6}\sum_{k=0}^{m-1}( \mathfrak {r}_{k+1}-\mathfrak {r}_{k})^{3}\max \bigl\{ \bigl\vert \mathcal {K}''(\mathfrak {r}_{k}) \bigr\vert , \bigl\vert \mathcal {K}''(\mathfrak {r}_{k+1}) \bigr\vert \bigr\} , \end{aligned} $$
By applying Proposition 20, the following consequence is obtained.
Corollary 21
Let
\(\mathcal {K}:\mathfrak {D}\subset [0,\infty )\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\)
such that
\(\mathcal {K}''\in L([\mathfrak {\alpha },\beta ])\), where
\(\mathfrak {\alpha },\beta \in \mathfrak {D}\)
with
\(\mathfrak {\alpha }<\beta \). If
\(|\mathcal {K}''|\)
is quasiconvex on
\([\mathfrak {\alpha },\beta ]\), then, for every partition
P
of
\([\mathfrak {\alpha },\beta ]\), we have:
I.
If
\(|\mathcal {K}''|\)
is increasing, then
$$ \bigl\vert \mathbb {E}(\mathcal {K},\mathbf{P})+5\mathbb {G}(\mathcal {K},\mathbf{P}) \bigr\vert \leq \frac{7}{6} \sum_{k=0}^{m-1}( \mathfrak {r}_{k+1}-\mathfrak {r}_{k})^{3} \bigl\vert \mathcal {K}''(\mathfrak {r}_{k+1}) \bigr\vert . $$
II.
If
\(|\mathcal {K}''|\)
is decreasing, then
$$ \bigl\vert \mathbb {E}(\mathcal {K},\mathbf{P})+5\mathbb {G}(\mathcal {K},\mathbf{P}) \bigr\vert \leq \frac{7}{6} \sum_{k=0}^{m-1}( \mathfrak {r}_{k+1}-\mathfrak {r}_{k})^{3} \bigl\vert \mathcal {K}''(\mathfrak {r}_{k}) \bigr\vert . $$
Proposition 22
Let
\(\mathcal {K}:\mathfrak {D}\subset [0,\infty )\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\)
such that
\(\mathcal {K}''\in L([\mathfrak {\alpha },\beta ])\), where
\(\mathfrak {\alpha },\beta \in \mathfrak {D}\)
with
\(\mathfrak {\alpha }<\beta \), and let
\(p>1\)
with
\(1/p+1/q=1\). If
\(|\mathcal {K}''|^{q}\)
is a quasiconvex function on
\([\mathfrak {\alpha },\beta ]\), then for every partition
P
of
\([\mathfrak {\alpha },\beta ]\)
$$ \begin{aligned} & \bigl\vert \mathbb {E}(\mathcal {K}, \mathbf{P})+5\mathbb {G}(\mathcal {K},\mathbf{P}) \bigr\vert \\ &\quad \leq \frac{1}{2} \biggl(\frac{2^{2p+1}-1}{2p+1} \biggr)^{\frac{1}{p}}\sum _{k=0}^{m-1}(\mathfrak {r}_{k+1}-\mathfrak {r}_{k})^{3} \bigl[\max \bigl\{ \bigl\vert \mathcal {K}''(\mathfrak {r}_{k}) \bigr\vert ^{q}, \bigl\vert \mathcal {K}''(\mathfrak {r}_{k+1}) \bigr\vert ^{q} \bigr\} \bigr]^{\frac{1}{q}}. \end{aligned} $$
Proof
Now applying Proposition 22, one obtains the following corollary.
Corollary 23
Let
\(\mathcal {K}:\mathfrak {D}\subset [0,\infty )\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\)
such that
\(\mathcal {K}''\in L([\mathfrak {\alpha },\beta ])\), where
\(\mathfrak {\alpha },\beta \in \mathfrak {D}\)
with
\(\mathfrak {\alpha }<\beta \), and let
\(p>1\)
with
\(1/p+1/q=1\). If
\(|\mathcal {K}''|^{q}\)
is a quasiconvex function on
\([\mathfrak {\alpha },\beta ]\), then, for every partition
P
of
\([\mathfrak {\alpha },\beta ]\), we get:
I.
If
\(|\mathcal {K}''|^{q}\)
is increasing, then
$$ \bigl\vert \mathbb {E}(\mathcal {K},\mathbf{P})+5\mathbb {G}(\mathcal {K},\mathbf{P}) \bigr\vert \leq \frac{1}{2} \biggl(\frac{2^{2p+1}-1}{2p+1} \biggr)^{\frac{1}{p}} \sum_{k=0}^{m-1}(\mathfrak {r}_{k+1}- \mathfrak {r}_{k})^{3} \bigl\vert \mathcal {K}''( \mathfrak {r}_{k+1}) \bigr\vert . $$
II.
If
\(|\mathcal {K}''|^{q}\)
is decreasing, then
$$ \bigl\vert \mathbb {E}(\mathcal {K},\mathbf{P})+5\mathbb {G}(\mathcal {K},\mathbf{P}) \bigr\vert \leq \frac{1}{2} \biggl(\frac{2^{2p+1}-1}{2p+1} \biggr)^{\frac{1}{p}} \sum_{k=0}^{m-1}(\mathfrak {r}_{k+1}- \mathfrak {r}_{k})^{3} \bigl\vert \mathcal {K}''( \mathfrak {r}_{k}) \bigr\vert . $$
Proposition 24
Let
\(\mathcal {K}:\mathfrak {D}\subset [0,\infty )\to \mathbb {R}\)
be a differentiable function on
\(\mathfrak {D}^{\circ }\)
such that
\(\mathcal {K}''\in L([\mathfrak {\alpha },\beta ])\), where
\(\mathfrak {\alpha },\beta \in \mathfrak {D}\)
with
\(\mathfrak {\alpha }<\beta \). If
\(|\mathcal {K}''|^{q}\)
is a quasiconvex function on
\([\mathfrak {\alpha },\beta ]\)
for
\(q\geq 1\), then for every partition
P
of
\([\mathfrak {\alpha },\beta ]\)
$$ \begin{aligned} & \bigl\vert \mathbb {E}(\mathcal {K}, \mathbf{P})+5\mathbb {G}(\mathcal {K},\mathbf{P}) \bigr\vert \leq \frac{7}{6} \sum_{k=0}^{m-1}(\mathfrak {r}_{k+1}- \mathfrak {r}_{k})^{3} \bigl[\max \bigl\{ \bigl\vert \mathcal {K}''(\mathfrak {r}_{k}) \bigr\vert ^{q}, \bigl\vert \mathcal {K}''(\mathfrak {r}_{k+1}) \bigr\vert ^{q} \bigr\} \bigr]^{\frac{1}{q}}. \end{aligned} $$
4 Applications to special means
In this section, we apply some of our theorems to the following special means for distinct real numbers x and y.
1.
Arithmetic mean:
$$ \mathcal {A}(x,y)=\frac{x+y}{2}. $$
2.
Geometric mean:
$$ \mathcal{G}(x,y)=\sqrt{xy}, \quad x,y>0. $$
3.
Harmonic mean:
$$ \mathcal{H}(x,y)=\frac{2xy}{x+y}. $$
4.
Logarithmic mean:
$$ \mathcal{L}(x,y)=\frac{x-y}{\ln \vert x \vert -\ln \vert y \vert }, \quad \vert x \vert \neq \vert y \vert , \text{ and } x, y\neq 0. $$
5.
Generalized logarithmic mean:
$$ \mathcal{L}_{m}(x,y)= \biggl[\frac{y^{m+1}-x^{m+1}}{(m+1)(y-x)} \biggr] ^{\frac{1}{m}}, \quad m\in \mathbb{N}. $$
Proposition 25
Suppose
\(\mathfrak {\alpha },\beta \in \mathbb {R}\)
with
\(\mathfrak {\alpha }<\beta \)
and
\(m\in \mathbb{N}\), \(m\geq 2\). Then we get
$$ \begin{aligned}[b] & \biggl\vert \mathcal{L}_{m}^{m}(\mathfrak {\alpha },\beta )-\mathcal {A}\bigl(\mathfrak {\alpha }^{m},\beta ^{m}\bigr)+\frac{5}{4}m(\beta - \mathfrak {\alpha }) \bigl(\beta ^{m-1}-\mathfrak {\alpha }^{m-1} \bigr) \biggr\vert \\ &\quad \leq \frac{7}{6}m(m-1) (\beta -\mathfrak {\alpha })^{2} \max \bigl\{ \vert \mathfrak {\alpha }\vert ^{m-2}, \vert \beta \vert ^{m-2} \bigr\} . \end{aligned} $$
(22)
Proof
Proposition 26
Suppose
\(\mathfrak {\alpha },\beta \in \mathbb {R}\)
with
\(\mathfrak {\alpha }<\beta \)
and
\(m\in \mathbb{N}\), \(m\geq 2\). Then, for
\(p,q>1\)
with
\(1/p+1/q=1\), we get
$$ \begin{aligned}[b] & \biggl\vert \mathcal{L}_{m}^{m}(\mathfrak {\alpha },\beta )-\mathcal {A}\bigl(\mathfrak {\alpha }^{m},\beta ^{m}\bigr)+\frac{5}{4}m(\beta - \mathfrak {\alpha }) \bigl(\beta ^{m-1}-\mathfrak {\alpha }^{m-1} \bigr) \biggr\vert \\ &\quad \leq \frac{(\beta -\mathfrak {\alpha })^{2}}{2}m(m-1) \biggl(\frac{2^{2p+1}-1}{2p+1} \biggr) ^{\frac{1}{p}} \bigl[\max \bigl\{ \vert \mathfrak {\alpha }\vert ^{q(m-2)}, \vert \beta \vert ^{q(m-2)} \bigr\} \bigr]^{\frac{1}{q}}. \end{aligned} $$
(23)
Proposition 27
Suppose
\(\mathfrak {\alpha },\beta \in \mathbb {R}\)
with
\(0<\mathfrak {\alpha }<\beta \). Then, for
\(q\geq 1\), we have
$$ \begin{aligned} \biggl\vert \mathcal{L}^{-1}( \mathfrak {\alpha },\beta )-\mathcal{H}^{-1}(\mathfrak {\alpha },\beta )+ \frac{5}{2}( \beta -\mathfrak {\alpha })^{2}\frac{\mathcal {A}(\mathfrak {\alpha },\beta )}{\mathcal{G}^{4}(\mathfrak {\alpha },\beta )} \biggr\vert \leq \frac{7}{6}(\beta -\mathfrak {\alpha })^{2}\frac{2^{1/q}}{\mathfrak {\alpha }^{3}}. \end{aligned} $$
(24)
Proof
Here, we consider the function \(\mathcal {K}(s)=1/s\), \(s\in [\mathfrak {\alpha },\beta ]\). The function \(|\mathcal {K}''(s)|^{q}= (\frac{2}{s^{3}} )^{q}\) is η-quasiconvex with respect to \(\eta (s,t)=s-t\). Employing Theorem 14 to \(\mathcal {K}\) and η, we get the desired result. □
5 Conclusion
Many integral inequalities associated with the η-quasiconvex and strongly η-quasiconvex functions have been established. Results obtained herein generalize, extend and refine some published theorems in the literature. Applications to the trapezoidal formula and some special means are also considered. We anticipate that this work will trigger further investigation in this direction. Some new inequalities can also be found in [3, 9, 10, 12, 14, 15, 20].
Acknowledgements
Many thanks go to the referees for their valuable comments and suggestions.
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