Some inequalities via fractional conformable integral operators
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- 01-12-2019
- Research
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Abstract
1 Introduction
The theory of fractional integral inequalities plays a vital role in the field of mathematical sciences. There is one of the most famous inequalities for convex functions known as Hermite–Hadamard inequality. Many researchers studied this inequality and published various generalizations and extensions by using fractional integral. We begin with the Hermite–Hadamard inequality, which is defined as follows: Let \(f:I\subseteq \mathbb{R}\rightarrow \mathbb{R}\) be a convex function and \(a, b\in I\) with \(a< b\), then
Further generalizations and extensions can be found in, e.g., [3, 8, 10, 17]. In [15], the Riemann–Liouville fractional integrals \(\mathfrak{I}_{a^{+}}^{\alpha }\) and \(\mathfrak{I}_{b^{-}}^{\alpha }\) of order \(\alpha >0\) are defined respectively by
and
where Γ is the gamma function (see [18]). In [7], the left- and right-sided fractional conformable integral operators are respectively defined by
and
where \(\beta \in \mathbb{C}\) and \(\Re (\beta )>0\). Obviously, if we consider \(a=0\), \(b=0\), and \(\alpha =1\) in (4) and (5), then we get the Riemann–Liouville fractional integrals (2) and (3) respectively. In [16], Set et al. defined the following one-sided conformable fractional integral operator:
Recently Rahman et al. [13, 14] established some new inequalities of the Grüss type and certain Chebyshev-type inequalities for conformable fractional integrals. In [5, 9, 11, 12], various researchers established generalized k-fractional conformable integral inequalities, Minkowski and Chebyshev type integral inequalities involving generalized k-fractional conformable integrals. The Hermite–Hadamard type inequalities for k-fractional conformable integrals are found in [6]. A significant contribution by Guessab and Schmeisser [4] is an investigation of sharp integral inequalities of the Hermite–Hadamard type.
$$ f \biggl(\frac{a+b}{2} \biggr)\leq \frac{1}{b-a} \int _{a}^{b}f(x)\,dx \leq \frac{f(a)+f(b)}{2}. $$
(1)
$$ \mathfrak{I}_{a^{+}}^{\alpha }f(x)= \frac{1}{\varGamma (\alpha )} \int _{a} ^{x} (x-t)^{\alpha -1}f(t)\,dt\quad \bigl(x>a, \Re (\alpha )>0 \bigr) $$
(2)
$$ \mathfrak{I}_{b^{-}}^{\alpha }f(x)= \frac{1}{\varGamma (\alpha )} \int _{x} ^{b} (t-x)^{\alpha -1}f(t)\,dt\quad \bigl(x< b, \Re (\alpha )>0 \bigr), $$
(3)
$$ ^{\beta }\mathfrak{I}_{a^{+}}^{\alpha }f(x)= \frac{1}{\varGamma (\beta )} \int _{a}^{x} \biggl(\frac{ (x-a )^{\alpha }- (t-a ) ^{\alpha }}{\alpha } \biggr)^{\beta -1}\frac{f(t)}{(t-a)^{1-\alpha }}\,dt,\quad x>a $$
(4)
$$ ^{\beta }\mathfrak{I}_{b^{-}}^{\alpha }f(x)= \frac{1}{\varGamma (\beta )} \int _{x}^{b} \biggl(\frac{ (b-x )^{\alpha }- (b-t ) ^{\alpha }}{\alpha } \biggr)^{\beta -1}\frac{f(t)}{(b-t)^{1-\alpha }}\,dt,\quad x< b, $$
(5)
$$ ^{\beta }\mathfrak{I}^{\alpha }f(x)= \frac{1}{\varGamma (\beta )} \int _{0} ^{x} \biggl(\frac{x^{\alpha }-\tau ^{\alpha }}{\alpha } \biggr)^{\beta -1}\frac{f(\tau )}{\tau ^{1-\alpha }}\,d\tau . $$
(6)
The paper is arranged as follows: In Sect. 2, the main results, which are reverse Minkowski and related Hermite–Hadamard type integral inequalities, are established by employing fractional conformable integral operators. The concluding remarks are given in Sect. 3.
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2 Main results
In this section, we use fractional conformable integral operator to develop reverse Minkowski and Hermite–Hadamard integral inequalities. The reverse Minkowski fractional integral inequality is presented in the following theorems.
Theorem 2.1
Let
\(\beta ,\alpha >0\), \(\sigma \geq 1\), and let
Φ, Ψ
be two positive functions on
\([0,\infty )\)
such that, for all
\(x>0\), \({}^{\beta }\mathfrak{I}^{\alpha }\varPhi ^{\sigma }(x)<\infty \), \({}^{\beta }\mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x)<\infty \). If
\(0< m\leq \frac{\varPhi (t)}{\varPsi (t)}\leq M\), \(t\in [0,x]\), then the following inequality holds:
$$\begin{aligned} \bigl({}^{\beta }\mathfrak{I}^{\alpha }\varPhi ^{\sigma }(x) \bigr)^{\frac{1}{ \sigma }}+ \bigl({}^{\beta } \mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x) \bigr)^{\frac{1}{\sigma }} \leq \frac{1+M(m+2)}{(m+1)(M+1)} \bigl({}^{ \beta }\mathfrak{I}^{\alpha }( \varPhi +\varPsi )^{\sigma }(x) \bigr)^{\frac{1}{ \sigma }}. \end{aligned}$$
(7)
Proof
Using the condition \(\frac{\varPhi (t)}{\varPsi (t)}< M\), \(t\in [0,x]\), \(x>0\), we have
Multiplying both sides of (8) by \(\frac{1}{\varGamma (\beta )} (\frac{x^{\alpha }-t^{\alpha }}{\alpha } )^{\beta -1}t^{\alpha -1}\) and integrating the resultant inequality with respect to t from 0 to x, we have
which can be written as
Hence, it follows that
Now, using the condition \(m\varPsi (t)\leq \varPhi (t)\), we have
which yields
Multiplying both sides of (10) by \(\frac{1}{\varGamma (\beta )} (\frac{x^{\alpha }-t^{\alpha }}{\alpha } )^{\beta -1}t^{\alpha -1}\) and integrating the resultant inequality with respect to t from 0 to x, we get
Thus, adding inequalities (9) and (11) yields the desired inequality. □
$$ (M+1)^{\sigma }\varPhi ^{\sigma }(t)\leq M^{\sigma } (\varPhi +\varPsi )^{\sigma }(t). $$
(8)
$$\begin{aligned} \frac{(M+1)^{\sigma }}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{ \alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\alpha -1}\varPhi ^{ \sigma }(t)\,dt &\leq \frac{M^{\sigma }}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\alpha -1} (\varPhi +\varPsi )^{\sigma }(t)\,dt, \end{aligned}$$
$$ ^{\beta }\mathfrak{I}^{\alpha }\varPhi ^{\sigma }(x)\leq \frac{M^{\sigma }}{(M+1)^{\sigma }}{}^{\beta }\mathfrak{I}^{\alpha } (\varPhi + \varPsi )^{\sigma }(x). $$
$$ \bigl({}^{\beta }\mathfrak{I}^{\alpha }\varPhi ^{\sigma }(x) \bigr)^{\frac{1}{ \sigma }}\leq \frac{M}{(M+1)} \bigl( {}^{\beta }\mathfrak{I}^{\alpha } (\varPhi +\varPsi )^{\sigma }(x) \bigr) ^{\frac{1}{\sigma }}. $$
(9)
$$ \biggl(1+\frac{1}{m} \biggr)\varPsi (t)\leq \frac{1}{m} \bigl(\varPhi (t)+\varPsi (t) \bigr), $$
$$ \biggl(1+\frac{1}{m} \biggr)^{\sigma }\varPsi ^{\sigma }(t)\leq \biggl(\frac{1}{m} \biggr)^{ \sigma } \bigl(\varPhi (t)+\varPsi (t) \bigr)^{\sigma }. $$
(10)
$$ \bigl({}^{\beta }\mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x) \bigr)^{\frac{1}{ \sigma }}\leq \frac{1}{(m+1)} \bigl( {}^{\beta }\mathfrak{I}^{\alpha } (\varPhi +\varPsi )^{\sigma }(x) \bigr) ^{\frac{1}{\sigma }}. $$
(11)
Theorem 2.2
Let
\(\beta , \alpha >0\), \(\beta \in \mathbb{C}\), \(\sigma \geq 1\), and let
Φ, Ψ
be two positive functions on
\([0,\infty )\)
such that, for all
\(x>0\), \({}^{\beta }\mathfrak{I}^{\alpha }\varPhi ^{\sigma }(x)< \infty \), \({}^{\beta }\mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x)<\infty \). If
\(0< m\leq \frac{\varPhi (t)}{\varPsi (t)}\leq M\), \(t\in [0,x]\), then the following inequality holds:
$$\begin{aligned} & \bigl({}^{\beta }\mathfrak{I}^{\alpha }\varPhi ^{\sigma }(x) \bigr)^{\frac{2}{ \sigma }}+ \bigl({}^{\beta } \mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x) \bigr)^{\frac{2}{\sigma }} \\ &\quad \geq \biggl(\frac{(M+1)(m+1)}{M}-2 \biggr) \bigl({}^{\beta } \mathfrak{I} ^{\alpha }\varPhi ^{\sigma }(x) \bigr)^{\frac{1}{\sigma }} \bigl({}^{\beta } \mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x) \bigr)^{\frac{1}{\sigma }}. \end{aligned}$$
(12)
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Proof
From the multiplication of inequalities (9) and (11), we have
Now, applying the Minkowski inequality to the right-hand side of (13), we obtain
Thus, from inequalities (13) and (14), we get the desired inequality (12). □
$$\begin{aligned} \biggl(\frac{(M+1)(m+1)}{M} \biggr) \bigl({}^{\beta } \mathfrak{I}^{\alpha } \varPhi ^{\sigma }(x) \bigr)^{\frac{1}{\sigma }} \bigl({}^{\beta } \mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x) \bigr)^{\frac{1}{\sigma }} \leq \bigl( \bigl[{}^{\beta }\mathfrak{I}^{\alpha } \bigl(\varPhi (x)+ \varPsi (x) \bigr)^{\sigma } \bigr]^{\frac{1}{\sigma }} \bigr)^{2}. \end{aligned}$$
(13)
$$\begin{aligned} &\bigl( \bigl[{}^{\beta }\mathfrak{I}^{\alpha } \bigl(\varPhi (x)+ \varPsi (x) \bigr)^{\sigma } \bigr]^{\frac{1}{\sigma }} \bigr)^{2} \\ &\quad \leq \bigl[ \bigl({} ^{\beta }\mathfrak{I}^{\alpha } \varPhi ^{\sigma }(x) \bigr)^{\frac{1}{ \sigma }}+ \bigl({}^{\beta } \mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x) \bigr)^{\frac{1}{\sigma }} \bigr]^{2} \\ &\quad \leq \bigl({}^{\beta }\mathfrak{I}^{\alpha }\varPhi ^{\sigma }(x) \bigr)^{\frac{1}{ \sigma }}+ \bigl({}^{\beta } \mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x) \bigr)^{\frac{1}{\sigma }}+2 \bigl({}^{\beta }\mathfrak{I}^{\alpha } \varPhi ^{\sigma }(x) \bigr)^{\frac{1}{\sigma }} \bigl({}^{\beta } \mathfrak{I}^{\alpha } \varPsi ^{\sigma }(x) \bigr)^{\frac{1}{\sigma }}. \end{aligned}$$
(14)
Lemma 2.3
([2])
Let
f
be a concave function on
\([a,b]\), then the following inequalities hold:
$$\begin{aligned} f(a)+f(b)\leq f(b+a-x)+f(x)\leq 2f \biggl(\frac{a+b}{2} \biggr). \end{aligned}$$
(15)
Theorem 2.4
Let
\(\beta , \alpha >0\), \(\beta \in \mathbb{C}\), \(r, s>1\), and let
Φ
and
Ψ
be two positive functions on
\([0,\infty )\). If
\(\varPhi ^{r}\)
and
\(\varPsi ^{s}\)
are two concave functions on
\([0,\infty )\), then the following inequality holds:
$$\begin{aligned} & 2^{-r-s} \bigl(\varPhi (0)+\varPhi \bigl(x^{\alpha } \bigr) \bigr)^{r} \bigl(\varPsi (0)+\varPsi \bigl(x ^{\alpha } \bigr) \bigr)^{s} \bigl({}^{\beta } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha } \bigr) \bigr)^{2} \\ &\quad \leq {}^{\beta }\mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha } \varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) {}^{\beta } \mathfrak{I}^{\alpha } \bigl(x ^{\alpha \beta -\alpha }\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr). \end{aligned}$$
(16)
Proof
Since the functions \(\varPhi ^{r}\) and \(\varPsi ^{s}\) are concave on \([0,\infty )\), therefore for any \(x>0\), \(\alpha >0\) and by Lemma 2.3, we have
and
Multiplying both sides of (17) and (18) by \(\frac{1}{\varGamma (\beta )} (\frac{x^{\alpha }-t^{\alpha }}{\alpha } )^{\beta -1}t^{\alpha \beta -1}\), \(t\in (0,x)\), and integrating the resultant inequalities from 0 to x, we get
and
Taking \(x^{\alpha }-t^{\alpha }=y^{\alpha }\), we have
and
Thus the use of (19) and (21) yields
Similarly, the use of (20) and (22) yields
From inequalities (23) and (24), it follows that
Since Φ and Ψ are positive functions, therefore for any \(x>0\), \(\alpha >0\), \(r\geq 1\), and \(s\geq 1\), we have
and
Hence, it follows that
and
From inequalities (28) and (29), we obtain
Thus, by combining (25) and (30), we get the desired result. □
$$\begin{aligned} \varPhi ^{r}(0)+\varPhi ^{r} \bigl(x^{\alpha } \bigr)\leq \varPhi ^{r} \bigl(x^{\alpha }-t^{\alpha } \bigr)+\varPhi ^{r} \bigl(t^{\alpha } \bigr)\leq 2\varPhi ^{r} \biggl(\frac{x^{\alpha }}{2} \biggr) \end{aligned}$$
(17)
$$\begin{aligned} \varPsi ^{s}(0)+\varPsi ^{s} \bigl(x^{\alpha } \bigr)\leq \varPsi ^{s} \bigl(x^{\alpha }-t^{\alpha } \bigr)+\varPhi ^{s} \bigl(t^{\alpha } \bigr)\leq 2\varPsi ^{s} \biggl(\frac{x^{\alpha }}{2} \biggr). \end{aligned}$$
(18)
$$\begin{aligned} &\frac{\varPhi ^{r}(0)+\varPhi ^{r}(x^{\alpha })}{\varGamma (\beta )} \int _{0} ^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{ \alpha \beta -1} \,dt \\ &\quad \leq \frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t ^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\alpha \beta -1}\varPhi ^{r} \bigl(x^{ \alpha }-t^{\alpha } \bigr)\,dt \\ &\qquad {}+\frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{ \alpha }}{\alpha } \biggr)^{\beta -1}t^{\alpha \beta -1}\varPhi ^{r} \bigl(t^{ \alpha } \bigr)\,dt \\ &\quad \leq \frac{2\varPhi ^{r}(\frac{x^{\alpha }}{2})}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t ^{\alpha \beta -1}\,dt \end{aligned}$$
(19)
$$\begin{aligned} &\frac{\varPsi ^{s}(0)+\varPsi ^{s}(x^{\alpha })}{\varGamma (\beta )} \int _{0} ^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{ \alpha \beta -1} \,dt \\ &\quad \leq \frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t ^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\alpha \beta -1}\varPsi ^{s} \bigl(x^{ \alpha }-t^{\alpha } \bigr)\,dt \\ &\qquad {}+\frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{ \alpha }}{\alpha } \biggr)^{\beta -1}t^{\alpha \beta -1}\varPsi ^{s} \bigl(t^{ \alpha } \bigr)\,dt \\ &\quad \leq \frac{2\varPsi ^{s}(\frac{x^{\alpha }}{2})}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t ^{\alpha \beta -1}\,dt. \end{aligned}$$
(20)
$$\begin{aligned} \frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\alpha \beta -1}\varPhi ^{r} \bigl(x^{\alpha }-t ^{\alpha } \bigr)\,dt ={}^{\beta } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) \end{aligned}$$
(21)
$$\begin{aligned} \frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\alpha \beta -1}\varPsi ^{s} \bigl(x^{\alpha }-t ^{\alpha } \bigr)\,dt ={}^{\beta } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr). \end{aligned}$$
(22)
$$\begin{aligned} & \bigl(\varPhi ^{r}(0)+\varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) \bigl(^{\beta } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha } \bigr) \bigr) \\ &\quad \leq 2 \bigl({}^{\beta }\mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta - \alpha }\varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) \bigr)\leq 2\varPhi ^{r} \biggl( \frac{x^{\alpha }}{2} \biggr) \bigl({}^{\beta }\mathfrak{I}^{\alpha } \bigl(x^{ \alpha \beta -\alpha } \bigr) \bigr). \end{aligned}$$
(23)
$$\begin{aligned} \bigl(\varPsi ^{s}(0)+\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr) \bigl(^{\beta }\mathfrak{I} ^{\alpha } \bigl(x^{\alpha \beta -\alpha } \bigr) \bigr) &\leq 2 \bigl( {}^{\beta } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr) \bigr) \\ & \leq 2\varPsi ^{s} \biggl(\frac{x^{\alpha }}{2} \biggr) \bigl( {}^{\beta }\mathfrak{I} ^{\alpha } \bigl(x^{\alpha \beta -\alpha } \bigr) \bigr). \end{aligned}$$
(24)
$$\begin{aligned} &\bigl(\varPhi ^{r}(0)+\varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) \bigl(\varPsi ^{s}(0)+\varPsi ^{s} \bigl(x ^{\alpha } \bigr) \bigr) \bigl({}^{\beta } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha } \bigr) \bigr)^{2} \\ &\quad \leq 4 \bigl({}^{\beta }\mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha } \varPhi ^{s} \bigl(x^{\alpha } \bigr) \bigr) \bigr) \bigl( {}^{\beta }\mathfrak{I}^{\alpha } \bigl(x ^{\alpha \beta -\alpha }\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr) \bigr). \end{aligned}$$
(25)
$$ \biggl(\frac{\varPhi ^{r}(0)+\varPhi ^{r}(x^{\alpha })}{2} \biggr)^{\frac{1}{r}} \geq 2^{-1} \bigl(\varPhi (0)+\varPhi \bigl(x^{\alpha } \bigr) \bigr) $$
(26)
$$ \biggl(\frac{\varPsi ^{s}(0)+\varPsi ^{s}(x^{\alpha })}{2} \biggr)^{\frac{1}{s}} \geq 2^{-1} \bigl(\varPsi (0)+\varPsi \bigl(x^{\alpha } \bigr) \bigr). $$
(27)
$$ \biggl(\frac{\varPhi ^{r}(0)+\varPhi ^{r}(x^{\alpha })}{2} \biggr) \bigl({}^{\beta } \mathfrak{I}^{\alpha } \bigl(t^{\alpha \beta -\alpha } \bigr) \bigr)\geq 2^{-r} \bigl(\varPhi (0)+\varPhi \bigl(x^{\alpha } \bigr) \bigr)^{r} \bigl({}^{\beta }\mathfrak{I} ^{\alpha } \bigl(t^{\alpha \beta -\alpha } \bigr) \bigr) $$
(28)
$$ \biggl(\frac{\varPsi ^{s}(0)+\varPsi ^{s}(x^{\alpha })}{2} \biggr) \bigl({}^{\beta } \mathfrak{I}^{\alpha } \bigl(t^{\alpha \beta -\alpha } \bigr) \bigr)\geq 2^{-s} \bigl(\varPsi (0)+\varPsi \bigl(x^{\alpha } \bigr) \bigr)^{s} \bigl({}^{\beta }\mathfrak{I} ^{\alpha } \bigl(t^{\alpha \beta -\alpha } \bigr) \bigr). $$
(29)
$$\begin{aligned} &\frac{ (\varPhi ^{r}(0)+\varPhi ^{r}(x^{\alpha }) ) (\varPsi ^{s}(0)+ \varPsi ^{s}(x^{\alpha }) )}{4} \bigl({}^{\beta } \mathfrak{I}^{\alpha } \bigl(t ^{\alpha \beta -\alpha } \bigr) \bigr)^{2} \\ &\quad \geq 2^{-r-s} \bigl(\varPhi (0)+\varPhi \bigl(x^{\alpha } \bigr) \bigr)^{r} \bigl(\varPsi (0)+ \varPsi \bigl(x^{\alpha } \bigr) \bigr)^{s} \bigl({}^{\beta }\mathfrak{I}^{\alpha } \bigl(t ^{\alpha \beta -\alpha } \bigr) \bigr)^{2}. \end{aligned}$$
(30)
Theorem 2.5
Let
\(\beta , \mu , \alpha >0\), \(\beta , \mu \in \mathbb{C}\), \(r>1\), \(s>1\), and let
Φ, Ψ
be two positive functions on
\([0,\infty )\). If
\(\varPhi ^{r}\)
and
\(\varPsi ^{s}\)
are two concave functions on
\([0,\infty )\), then we have the following inequality:
$$\begin{aligned} &2^{2-r-s} \bigl(\varPhi (0)+\varPhi (x) \bigr)^{r} \bigl(\varPsi (0)+\varPsi (x) \bigr)^{s} \bigl({}^{\beta }\mathfrak{I}^{\alpha } \bigl(x^{\alpha \mu -\alpha } \bigr) \bigr)^{2} \\ &\quad \leq \biggl[\frac{\varGamma (\mu )}{\varGamma (\beta )}{}^{\mu } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPhi ^{r} \bigl(x^{ \alpha } \bigr) \bigr)+{}^{\beta }\mathfrak{I}^{\alpha } \bigl(x^{\mu \alpha - \alpha } \varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) \biggr] \\ &\qquad {}\times \biggl[\frac{\varGamma (\mu )}{\varGamma (\beta )}{}^{\mu } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPsi ^{s} \bigl(x^{ \alpha } \bigr) \bigr)+{}^{\beta }\mathfrak{I}^{\alpha } \bigl(x^{\mu \alpha - \alpha } \varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr) \biggr]. \end{aligned}$$
(31)
Proof
Multiplying both sides of inequalities (17) and (18) by \(\frac{1}{\varGamma (\beta )} (\frac{x^{\alpha }-t ^{\alpha }}{\alpha } )^{\beta -1}t^{\mu \alpha -1}\), \(t\in (0,x)\) and then integrating the resultant inequalities with respect to t from 0 to x, we have
and
Now, using \(x^{\alpha }-t^{\alpha }=y^{\alpha }\), we have
and
Thus, from (32) and (34), we can write
Similarly, from inequalities (33) and (35), we obtain
From (36) and (37), it follows that
Since Φ and Ψ are positive functions, therefore for any \(x>0\), \(\alpha >0\), \(r\geq 1\), \(s\geq 1\), we have
and
Thus from (39) and (40) it follows that
Combining inequalities (38) and (41), we get the desired proof. □
$$\begin{aligned} &\frac{\varPhi ^{r}(0)+\varPhi ^{r}(x^{\alpha })}{\varGamma (\beta )} \int _{0} ^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{ \mu \alpha -1}\,dt \\ &\quad \leq \frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t ^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\mu \alpha -1} \varPhi ^{r} \bigl(x^{ \alpha }-t^{\alpha } \bigr)\,dt \\ &\qquad {}+\frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x ^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\mu \alpha -1} \varPhi ^{r} \bigl(t^{\alpha } \bigr)\,dt \\ &\quad \leq \frac{2\varPhi ^{r}(\frac{x^{\alpha }}{2})}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t ^{\mu \alpha -1}\,dt \end{aligned}$$
(32)
$$\begin{aligned} &\frac{\varPsi ^{s}(0)+\varPsi ^{s}(x^{\alpha })}{\varGamma (\beta )} \int _{0} ^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{ \mu \alpha -1}\,dt \\ &\quad \leq \frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t ^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\mu \alpha -1} \varPsi ^{s} \bigl(x^{ \alpha }-t^{\alpha } \bigr)\,dt \\ &\qquad {}+\frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x ^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\mu \alpha -1} \varPsi ^{s} \bigl(t^{\alpha } \bigr)\,dt \\ &\quad \leq \frac{2\varPsi ^{s}(\frac{x^{\alpha }}{2})}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t ^{\mu \alpha -1}\,dt. \end{aligned}$$
(33)
$$ \frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\mu \alpha -1} \varPhi ^{r} \bigl(x^{\alpha }-t ^{\alpha } \bigr)\,dt =\frac{\varGamma (\mu )}{\varGamma (\beta )} {}^{\mu } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPhi ^{r} \bigl(x^{ \alpha } \bigr) \bigr) $$
(34)
$$ \frac{1}{\varGamma (\beta )} \int _{0}^{x} \biggl(\frac{x^{\alpha }-t^{\alpha }}{\alpha } \biggr)^{\beta -1}t^{\mu \alpha -1} \varPsi ^{s} \bigl(x^{\alpha }-t ^{\alpha } \bigr)\,dt =\frac{\varGamma (\mu )}{\varGamma (\beta )} {}^{\mu } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPsi ^{s} \bigl(x^{ \alpha } \bigr) \bigr). $$
(35)
$$\begin{aligned} \bigl(\varPhi ^{r}(0)+\varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr){}^{\beta }\mathfrak{I} ^{\alpha } \bigl(x^{\alpha \mu -\alpha } \bigr) &\leq \frac{\varGamma ( \mu )}{\varGamma (\beta )}{} ^{\mu }\mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) +{}^{\beta } \mathfrak{I} ^{\alpha } \bigl(x^{\alpha \mu -\alpha }\varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) \\ &\leq 2\varPhi ^{r} \biggl(\frac{x^{\alpha }}{2} \biggr) {}^{\beta }\mathfrak{I}^{ \alpha } \bigl(x^{\alpha \mu -\alpha } \bigr). \end{aligned}$$
(36)
$$\begin{aligned} \bigl(\varPsi ^{s}(0)+\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr){}^{\beta }\mathfrak{I} ^{\alpha } \bigl(x^{\alpha \mu -\alpha } \bigr) &\leq \frac{\varGamma ( \mu )}{\varGamma (\beta )}{} ^{\mu }\mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr) +{}^{\beta } \mathfrak{I} ^{\alpha } \bigl(x^{\alpha \mu -\alpha }\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr) \\ &\leq 2\varPsi ^{s} \biggl(\frac{x^{\alpha }}{2} \biggr) {}^{\beta }\mathfrak{I}^{ \alpha } \bigl(x^{\alpha \mu -\alpha } \bigr). \end{aligned}$$
(37)
$$\begin{aligned} & \bigl(\varPhi ^{r}(0)+\varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) \bigl(\varPsi ^{s}(0)+\varPsi ^{s} \bigl(x ^{\alpha } \bigr) \bigr) \bigl({}^{\beta } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \mu -\alpha } \bigr) \bigr)^{2} \\ &\quad \leq \biggl[\frac{\varGamma (\mu )}{\varGamma (\beta )}{}^{\mu } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPhi ^{r} \bigl(x^{ \alpha } \bigr) \bigr) +{}^{\beta }\mathfrak{I}^{\alpha } \bigl(x^{\alpha \mu -\alpha } \varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr) \biggr] \\ & \qquad {}\times\biggl[\frac{\varGamma (\mu )}{\varGamma (\beta )}{}^{\mu }\mathfrak{I} ^{\alpha } \bigl(x^{\alpha \beta -\alpha }\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr) + {}^{\beta }\mathfrak{I}^{\alpha } \bigl(x^{\alpha \mu -\alpha } \varPsi ^{s} \bigl(x ^{\alpha } \bigr) \bigr) \biggr]. \end{aligned}$$
(38)
$$\begin{aligned} &\frac{\varPhi ^{r}(0)+\varPhi ^{r}(x^{\alpha })}{2}{}^{\beta }\mathfrak{I} ^{\alpha } \bigl(x^{\alpha \mu -\alpha } \bigr)\geq 2^{-r} \bigl( \varPhi ^{r}(0)+ \varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr)^{r}{}^{\beta }\mathfrak{I}^{\alpha } \bigl(x ^{\alpha \mu -\alpha } \bigr) \end{aligned}$$
(39)
$$\begin{aligned} &\frac{\varPsi ^{s}(0)+\varPsi ^{s}(x^{\alpha })}{2}{}^{\beta }\mathfrak{I} ^{\alpha } \bigl(x^{\alpha \mu -\alpha } \bigr)\geq 2^{-s} \bigl( \varPsi ^{s}(0)+ \varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr)^{s}{}^{\beta }\mathfrak{I}^{\alpha } \bigl(x ^{\alpha \mu -\alpha } \bigr). \end{aligned}$$
(40)
$$\begin{aligned} &\frac{ (\varPhi ^{r}(0)+\varPhi ^{r}(x^{\alpha }) ) (\varPsi ^{s}(0)+ \varPsi ^{s}(x^{\alpha }) )}{4} \bigl[ {}^{\beta } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \mu -\alpha } \bigr) \bigr]^{2} \\ &\quad \geq 2^{-r-s} \bigl(\varPhi ^{r}(0)+\varPhi ^{r} \bigl(x^{\alpha } \bigr) \bigr)^{r} \bigl( \varPsi ^{s}(0)+\varPsi ^{s} \bigl(x^{\alpha } \bigr) \bigr)^{s} \bigl[{}^{\beta } \mathfrak{I}^{\alpha } \bigl(x^{\alpha \mu -\alpha } \bigr) \bigr]^{2}. \end{aligned}$$
(41)
3 Concluding remarks
In this paper, we established Minkowski and Hermite–Hadamard inequalities for conformable fractional integral operator. If we consider \(\alpha =1\) throughout the paper, then the obtained results will reduce to the said inequalities obtained by Dahmani [2]. Similarly, if we consider \(\alpha =\beta =1\), then all the results will lead to the classical inequalities obtained by [1].
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