Some Reverse Hölder Type Inequalities Involving $(k,s)-$Riemann-Liouville Fractional Integrals

Abstract

In this paper, we aim to present the improved version of the reverse Hölder type inequalities by taking $(k,s)-$Riemann-Liouville fractional integrals. Furthermore, we also discuss some applications of Theorem 1 using some types of fractional integrals.

1 Introduction

Fractional integral inequalities involving $(k,s)-$ type integrals attract the attentions of many researchers due their diverse applications see, for examples, [1,2,3,4]. In [5], Farid et al. an integral inequality obtained by Mitrinovic and Pecaric was generalized to measure space as follows.

Theorem 1.

Let $(\varOmega _1,\varSigma _1,\mu _1)$,$(\varOmega _2,\varSigma _2,\mu _2)$ be measure spaces with $\sigma -$finite measures and let $f_i:\varOmega _2 \rightarrow \mathbb {R}$, $i=1,2,3,4$ be non-negative functions. Let g be the function having representation

$$g(x)=\int _{\varOmega _1}k(x,t)f(t)d\mu _1(t),$$

where $k:\varOmega _2\times \varOmega _1\rightarrow \mathbb {R}$ is a general non-negative kernel and $f:\varOmega _1\rightarrow \mathbb {R}$ is real-valued function, and $\mu _2$ is a non-decreasing function. If p, q are two real numbers such that $\frac{1}{p}+\frac{1}{q}=1$, $p>1$, then

$$\begin{aligned}&\int _{\varOmega _2}f_1(x)f_2(x)g(x)d\mu _2(x) \\\le & {} C\left( \int _{\varOmega _2}f_3(x)g(x)d\mu _2(x)\right) ^\frac{1}{p} \left( \int _{\varOmega _2}f_4(x)g(x)d\mu _2(x)\right) ^\frac{1}{q}, \nonumber \end{aligned}$$

(1)

where

$$\begin{aligned}&C=\sup _{t\in \varOmega _1}\Bigg \{ \left( \int _a^b \ k(x,t)f_1(x)f_2(x)d\mu _2(x)\right) \\&\left( \int _a^b \ k(x,t)f_3(x)d\mu _2(x)\right) ^\frac{-1}{p} \left( \int _a^b \ k(x,t)f_4(x)d\mu _2(x)\right) ^\frac{-1}{q}\Bigg \}.\nonumber \end{aligned}$$

(2)

The following definitions and results are also required.

2 Preliminaries

Recently fractional integral inequalities are considered to be an important tool of applied mathematics and their many applications described by a number of researchers. As well as, the theory of fractional calculus is used in solving differential, integral and integro-differential equations and also in various other problems involving special functions [6,7,8].

We begin by recalling the well-known results.

1.:

The Pochhammer k-symbol $(x)_{n,k}$ and the k-gamma function $\varGamma _k$ are defined as follows (see [9]):

$$\begin{aligned} (x)_{n,k}:=x(x+k)(x+2k)\cdots \left( x+(n-1)k\right) \quad \left( n \in \mathbb {N};\, k >0\right) \end{aligned}$$

(3)

and

$$\begin{aligned} \varGamma _k(x):= \lim _{n \rightarrow \infty }\, \frac{n!\,k^n\, (nk)^{\frac{x}{k}-1}}{(x)_{n,k}} \quad \left( k >0;\, x \in \mathbb {C}{\setminus } k \mathbb {Z}_0^-\right) , \end{aligned}$$

(4)

where $k \mathbb {Z}_0^-:=\left\{ kn\, :\, n \in \mathbb {Z}_0^- \right\}$. It is noted that the case $k=1$ of equation ((3)) and equation ((4)) reduces to the familiar Pochhammer symbol $(x)_{n}$ and the gamma function $\varGamma$. The function $\varGamma _k$ is given by the following integral:

$$\begin{aligned} \varGamma _k(x)= \int _0^\infty \, t^{x-1}\, e^{-\frac{t^k}{k}}\,dt \quad (\mathfrak {R}(x)>0). \end{aligned}$$

(5)

The function $\varGamma _k$ defined on $\mathbb {R}^+$ is characterized by the following three properties: (i) $\varGamma _k(x+k)=x\, \varGamma _k(x)$; (ii) $\varGamma _k(k)=1$; (iii) $\varGamma _k(x)$ is logarithmically convex. It is easy to see that

$$\begin{aligned} \varGamma _k(x)= k^{\frac{x}{k}-1} \, \varGamma \left( \frac{x}{k}\right) \quad \left( \mathfrak {R}(x)>0;\, k >0 \right) . \end{aligned}$$

(6)

2.:

Mubeen and Habibullah [10] introduced k-fractional integral of the Riemann-Liouville type of order $\alpha$ as follows:

$$\begin{aligned} _{k}J^{\alpha }_{a}\left[ f\left( t\right) \right] =\frac{1}{\varGamma _{k}(\alpha ) }\int _{a}^{t}\left( t-\tau \right) ^{\frac{\alpha }{k}-1}f\left( \tau \right) d\tau ,\left( \alpha>0,x>0,k>0\right) , \end{aligned}$$

(7)

which, upon setting $k=1$, is seen to yield the classical Riemann-Liouville fractional integral of order $\alpha$:

$$\begin{aligned} J^{\alpha }_{a}\left\{ f(t)\right\} := \,_{1}J^{\alpha }_{a} \left\{ f(t)\right\} =\frac{1}{\varGamma (\alpha )} \int _a^t\,(t-\tau )^{\alpha -1}f(\tau )\,d\tau \quad \left( \alpha>0;\, t>a\right) . \end{aligned}$$

(8)

3.:

Sarikaya et al. [11] presented (k, s)-fractional integral of the Riemann-Liouville type of order $\alpha$, which is a generalization of the k-fractional integral (7), defined as follows:

$$\begin{aligned} {}_{k}^{s}J_{a}^{\alpha }\left[ f\left( t\right) \right] :=\frac{\left( s+1\right) ^{1-\frac{\alpha }{k}}}{k\varGamma _{k}\left( \alpha \right) } \int _{a}^{t}\left( t^{s+1}-\tau ^{s+1}\right) ^{\frac{\alpha }{k}-1}\tau ^{s}f\left( \tau \right) d\tau ,\ \tau \in \left[ a,b\right] , \end{aligned}$$

(9)

where $k>0,s\in \mathbb {R} \backslash \left\{ -1\right\}$ and which, upon setting $s=0$, immediately reduces to the k-integral (7).

4.:

In [11], the following results have been obtained. For $f\$be continuous on $[a,b],\ k>0$ and $s\in \mathbb {R}{\setminus }\{-1\}$. Then,

$$\begin{aligned} {}_{k}^{s}J_{a}^{\alpha }\left[ _{k}^{s}J_{a}^{\beta }f\left( t\right) \right] =\ _{k}^{s}J_{a}^{\alpha +\beta }f\left( t\right) =\ _{k}^{s}J_{a}^{\beta } \left[ _{k}^{s}J_{a}^{\alpha }f\left( t\right) \right] , \end{aligned}$$

(10)

and

$$\begin{aligned} _{k}^{s}J_{a}^{\alpha }\left[ \left( x^{s+1}-a^{s+1}\right) ^{\frac{\beta }{k }-1}\right] =\frac{\varGamma _{k}(\beta )}{\left( s+1\right) ^{\frac{\alpha }{k} }\varGamma _{k}(\alpha +\beta )}\left( x^{s+1}-a^{s+1}\right) ^{\frac{\alpha +\beta }{k}-1}, \end{aligned}$$

for all $\alpha ,\beta >0,\ x\in \left[ a,b\right]$ and $\varGamma _{k}\$denotes the $k-$gamma function.

5.:

Also, in [12], Akkurt et al. introduced $(k,H)-$fractional integral. Let (a, b) be a finite interval of the real line $\mathbb {R}$ and $\mathfrak {R}(\alpha )> 0$. Also let h(x) be an increasing and positive monotone function on (a, b], having a continuous derivative $h'(x)$ on (a, b). The left- and right-sided fractional integrals of a function f with respect to another function h on [a, b] are defined by

$$\begin{aligned}&\left( _{k}J_{a^+,h}^{\alpha }f\right) (x) \\:= & {} \frac{1}{k\varGamma _k(\alpha )} \int _a^x[h(x)-h(t)]^{\frac{\alpha }{k}-1}h'(t)f(t)dt, \ k>0 \ , \mathfrak {R}(\alpha )> 0 \nonumber \end{aligned}$$

(11)

$$\begin{aligned}&\left( _{k}J_{b^-,h}^{\alpha }f\right) (x) \\:= & {} \frac{1}{k\varGamma _k(\alpha )} \int _x^b[h(x)-h(t)]^{\frac{\alpha }{k}-1}h'(t)f(t)dt, \ k>0 \ , \mathfrak {R}(\alpha )> 0. \nonumber \end{aligned}$$

(12)

Recently, Tomar and Agarwal [13] obtained following results for $(k,s)-$fractional integrals.

Theorem 2

(Hölder Inequality for (k, s)-fractional integrals). Let $f,g:[a,b]\rightarrow \mathbb {R}$ be continuous functions and $p,q>0$ with $\frac{1}{p}+\frac{1}{q}=1$. Then, for all $t>0,\ k>0,\ \alpha >0,s\in \mathbb {R}-\{-1\},$

$$\begin{aligned}&_{k}^{s}J_{a}^{\alpha }\left| \ fg\left( t\right) \right| \le \left[ _{k}^{s}J_{a}^{\alpha }\left| f\left( t\right) \right| ^p\right] ^\frac{1}{p} \left[ _{k}^{s}J_{a}^{\alpha }\left| g\left( t\right) \right| ^q\right] ^\frac{1}{q}\text {.} \end{aligned}$$

(13)

Lemma 1.

Let $f,g:[a,b]\rightarrow \mathbb {R}$ be two positive functions and $\frac{1}{p}+\frac{1}{q}=1$, $\alpha ,k>0$ and $s\in \mathbb {R}-\{-1\}$, such that for $t\in [a,b]$, $_{k}^{s}J_{a}^{\alpha }f^p(t)< \infty$, $_{k}^{s}J_{a}^{\alpha }g^q(t)< \infty$. If

$$\begin{aligned}&0\le m\le \frac{f(\tau )}{g(\tau )}\le M <\infty , \tau \in [a,b], \end{aligned}$$

(14)

then the inequality

$$\begin{aligned}&\left[ _{k}^{s}J_{a}^{\alpha }f(t)\right] ^\frac{1}{p} \left[ _{k}^{s}J_{a}^{\alpha }g(t)\right] ^\frac{1}{q} \le \left( \frac{M}{m}\right) ^\frac{1}{pq} \ _{k}^{s}J_{a}^{\alpha } \left[ f^\frac{1}{p}(t)g^\frac{1}{q}(t)\right] \end{aligned}$$

(15)

holds.

Lemma 2.

Let $f,g:[a,b]\rightarrow \mathbb {R}$ be two positive functions $\alpha ,k>0$ and $s\in \mathbb {R}-\{-1\}$, such that for $t\in [a,b]$, $_{k}^{s}J_{a}^{\alpha }f^p(t)< \infty$, $_{k}^{s}J_{a}^{\alpha }g^q(t)< \infty$. If

$$\begin{aligned}&0\le m\le \frac{f^p(\tau )}{g^q(\tau )}\le M <\infty , \tau \in [a,b], \end{aligned}$$

(16)

then we have

$$\begin{aligned}&\left[ _{k}^{s}J_{a}^{\alpha }f^p(t)\right] ^\frac{1}{p} \left[ _{k}^{s}J_{a}^{\alpha }g^q(t)\right] ^\frac{1}{q} \le \left( \frac{M}{m}\right) ^\frac{1}{pq} \ _{k}^{s}J_{a}^{\alpha }\left( f(t)g(t)\right) , \end{aligned}$$

(17)

where $p>1$ and $\frac{1}{p}+\frac{1}{q}=1$.

Motivated by this work, we establish in this paper some new extensions of the reverse Hölder type inequalities by taking $(k,s)-$Riemann-Liouville fractional integrals .

3 Reverse Hölder Type Inequalites

In this section we prove our main results (Theorems 3 and 4).

Theorem 3.

Let f(x) and g(x) be integrable functions and let $0<p<1$, $\frac{1}{p}+\frac{1}{q}=1$. Then, the following inequality holds

$$\begin{aligned} _{k}^{s}J_{a}^{\alpha }\left| fg(t) \right|\ge & {} _{k}^{s}J_{a}^{\alpha }\left| f^p(t) \right| ^\frac{1}{p} \ _{k}^{s}J_{a}^{\alpha }\left| f^q(t) \right| ^\frac{1}{q}. \end{aligned}$$

(18)

Proof.

Set $c=\frac{1}{p}$, $q=-pd$. Then we have $d=\frac{c}{c-1}.$ By the Hölder inequality for $(k,s)-$fractional integrals, we have

$$\begin{aligned} {}_{k}^{s}J_{a}^{\alpha }\left| f^p(t) \right|= & {} _{k}^{s}J_{a}^{\alpha }\left| fg(t) \right| ^p \left| g^{-p}(t)\right| \nonumber \\\le & {} \left[ _{k}^{s}J_{a}^{\alpha }\left| fg(t) \right| ^{pc}\right] ^\frac{1}{c} \left[ _{k}^{s}J_{a}^{\alpha }\left| g(t) \right| ^{-pd}\right] ^\frac{1}{d} \nonumber \\= & {} \left[ _{k}^{s}J_{a}^{\alpha }\left| fg(t) \right| \right] ^\frac{1}{c} \left[ _{k}^{s}J_{a}^{\alpha }\left| g(t) \right| ^{q}\right] ^{1-p}. \end{aligned}$$

(19)

In equation (19), multiplying both sides by $\left( _{k}^{s}J_{a}^{\alpha }\left| g^{q}(t) \right| \right) ^{p-1}$, we obtain

$$\begin{aligned}&_{k}^{s}J_{a}^{\alpha }\left| f^p(t) \right| \left( _{k}^{s}J_{a}^{\alpha }\left| g^q(t) \right| \right) ^{p-1} \nonumber \\\le & {} \left[ _{k}^{s}J_{a}^{\alpha }\left| fg(t) \right| \right] ^p. \end{aligned}$$

(20)

Inequality (20) implies inequality

$$\begin{aligned} _{k}^{s}J_{a}^{\alpha }\left| fg(t) \right|\ge & {} _{k}^{s}J_{a}^{\alpha }\left| f^p(t) \right| ^\frac{1}{p} \ _{k}^{s}J_{a}^{\alpha }\left| f^q(t) \right| ^\frac{1}{q} \end{aligned}$$

(21)

which completes this theorem.

Theorem 4.

Suppose $p,q,l>0$ and $\frac{1}{p}+\frac{1}{q}+\frac{1}{l}=1$. If f, g and h are positive functions such that

• i.) $0<m\le \frac{f^\frac{p}{s}}{g^\frac{g}{s}}\le M<\infty$ for some $l>0$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{s}$,

• ii.) $0<m\le \frac{(fg)^s}{h^r}\le M<\infty ,$

then

$$\begin{aligned}&\left( _{k}^{s}J_{a}^{\alpha } f^p(t)\right) ^\frac{1}{p} \left( _{k}^{s}J_{a}^{\alpha } f^q(t)\right) ^\frac{1}{q} \left( _{k}^{s}J_{a}^{\alpha } f^r(t)\right) ^\frac{1}{r} \nonumber \\\le & {} \left( \frac{M}{m}\right) ^{ \frac{1}{sr}+\frac{pq}{s^3} } \ _{k}^{s}J_{a}^{\alpha }(fgh)(t). \end{aligned}$$

(22)

Proof.

Let $\frac{1}{p}+\frac{1}{q}=\frac{1}{s}$ for some $s>0$. Thus, $\frac{s}{p}+\frac{s}{q}=1$ and $\frac{1}{s}+\frac{1}{r}=1$. If we use ii and Lemma 2 for $H=fg$ and h, then we get

$$\begin{aligned}&\left( _{k}^{s}J_{a}^{\alpha }H^s(t)\right) ^\frac{1}{s} \left( _{k}^{s}J_{a}^{\alpha }h^r(t)\right) ^\frac{1}{r} \le \left( \frac{M}{m}\right) ^{\frac{1}{sr}} \left( _{k}^{s}J_{a}^{\alpha }(Hh)(t)\right) \end{aligned}$$

(23)

which is equivalent to

$$\begin{aligned}&\left( _{k}^{s}J_{a}^{\alpha }[f^s(t)g^s(t)]\right) ^\frac{1}{s} \left( _{k}^{s}J_{a}^{\alpha }h^r(t)\right) ^\frac{1}{r} \le \left( \frac{M}{m}\right) ^{\frac{1}{sr}} \left( _{k}^{s}J_{a}^{\alpha }(fgh)(t)\right) . \end{aligned}$$

(24)

Now, using i and the fact that $\frac{s}{p}+\frac{s}{q}=1$, and applying Lemma 2 to $f^s$ and $g^s$, we also have

$$\begin{aligned}&\left( _{k}^{s}J_{a}^{\alpha }f^p(t)\right) ^\frac{s}{p} \left( _{k}^{s}J_{a}^{\alpha }g^q(t)\right) ^\frac{s}{q} \le \left( \frac{M}{m}\right) ^{\frac{pq}{s^2}} \left( _{k}^{s}J_{a}^{\alpha }f^s(t)g^s(t)\right) \end{aligned}$$

(25)

which is equivalent to

$$\begin{aligned}&\left( _{k}^{s}J_{a}^{\alpha }f^p(t)\right) ^\frac{1}{p} \left( _{k}^{s}J_{a}^{\alpha }g^q(t)\right) ^\frac{1}{q} \le \left( \frac{M}{m}\right) ^{\frac{pq}{s^3}} \left( _{k}^{s}J_{a}^{\alpha }f^s(t)g^s(t)\right) ^\frac{1}{s}. \end{aligned}$$

(26)

Combining equations (24) and (26), we obtain desired inequality equation (22), which is complete the proof.

4 Applications for Some Types Fractional Integrals

Here in this section, we discuss some applications of Theorem 1 in the terms of Theorems 5-7 and Corollary 1-5.

Theorem 5.

Let p, q be two real numbers such that $\frac{1}{p}+\frac{1}{q}=1$, $p>1$ and let $f\$be continuous on $[a,b],\ k>0$ and $s\in \mathbb {R}{\setminus }\{-1\}$ . Then

$$\begin{aligned}&\int _a^bf_1(x)f_2(x)_{k}^sJ_{a}^{\alpha }f(x)dx \\\le & {} C\left( \int _a^bf_3(x)_{k}^sJ_{a}^{\alpha }f(x)dx\right) ^\frac{1}{p} \left( \int _a^bf_4(x)_{k}^sJ_{a}^{\alpha }f(x)dx\right) ^\frac{1}{q}, \nonumber \end{aligned}$$

(27)

where

$$\begin{aligned}&C=\sup _{t\in [a,b]}\Bigg \{ \left( \int _a^b \left( x^{s+1}-t^{s+1}\right) ^{\frac{\alpha }{k}-1}f_1(x)f_2(x)dx\right) \\&\left( \int _a^b \left( x^{s+1}-t^{s+1}\right) ^{\frac{\alpha }{k}-1}f_3(x)dx\right) ^\frac{-1}{p} \left( \int _a^b \left( x^{s+1}-t^{s+1}\right) ^{\frac{\alpha }{k}-1} f_4(x)dx\right) ^\frac{-1}{q}\Bigg \}.\nonumber \end{aligned}$$

(28)

Proof.

In Theorem 1, if we take $\varOmega _1=\varOmega _2=(a,b)$, $d\mu _1(t)=dt, d\mu _2(x)=dx$ and the kernel

$$ k(x,t)={\left\{ \begin{array}{ll} \frac{\left( s+1\right) ^{1-\frac{\alpha }{k}} \left( t^{s+1}-\tau ^{s+1}\right) ^{\frac{\alpha }{k}-1}\tau ^{s}}{k\varGamma _{k}(\alpha )} &{} \text { if } a\le t\le x \\ 0 &{} \text { if } x<t\le b, \end{array}\right. }$$

then g(x) becomes $\ _{k}^{s}J_{a}^{\alpha }f(t)$ and so we get desired inequality (27). This completes the proof of Theorem 5.

Corollary 1.

In Theorem 5, if we take $s=0$, then we get

$$\begin{aligned}&\int _a^bf_1(x)f_2(x)_{k}J_{a}^{\alpha }f(x)dx \\\le & {} C\left( \int _a^bf_3(x)_{k}J_{a}^{\alpha }f(x)dx\right) ^\frac{1}{p} \left( \int _a^bf_4(x)_{k}J_{a}^{\alpha }f(x)dx\right) ^\frac{1}{q}, \nonumber \end{aligned}$$

(29)

where

$$\begin{aligned}&C=\sup _{t\in [a,b]}\Bigg \{ \left( \int _a^b \left( x-t\right) ^{\frac{\alpha }{k}-1}f_1(x)f_2(x)dx\right) \\&\left( \int _a^b \left( x-t\right) ^{\frac{\alpha }{k}-1}f_3(x)dx\right) ^\frac{-1}{p} \left( \int _a^b \left( x-t\right) ^{\frac{\alpha }{k}-1} f_4(x)dx\right) ^\frac{-1}{q}\Bigg \}.\nonumber \end{aligned}$$

(30)

Remark 1.

In Corollary 1, $\alpha =k=1$, Theorem 1 reduces to Theorem 3.1 in [5].

Corollary 2.

In Theorem 5, if we take $f_3(x)=f_1^p(x)$ and $f_4(x)=f_2^q(x)$, then we get

$$\begin{aligned}&\int _a^bf_1(x)f_2(x)_{k}^{s}J_{a}^{\alpha }f(x)dx \\\le & {} C\left( \int _a^bf_1^p(x)_{k}^{s}J_{a}^{\alpha }f(x)dx\right) ^\frac{1}{p} \left( \int _a^bf_2^q(x)_{k}^{s}J_{a}^{\alpha }f(x)dx\right) ^\frac{1}{q}, \nonumber \end{aligned}$$

(31)

where

$$\begin{aligned}&C=\sup _{t\in [a,b]}\Bigg \{ \left( \int _a^b \left( x^{s+1}-t^{s+1}\right) ^{\frac{\alpha }{k}-1}f_1(x)f_2(x)dx\right) \\&\left( \int _a^b \left( x^{s+1}-t^{s+1}\right) ^{\frac{\alpha }{k}-1}f_1^p(x)dx\right) ^\frac{-1}{p} \left( \int _a^b \left( x^{s+1}-t^{s+1}\right) ^{\frac{\alpha }{k}-1} f_2^q(x)dx\right) ^\frac{-1}{q}\Bigg \}.\nonumber \end{aligned}$$

(32)

Corollary 3.

In Corollary 2, if we take $s=0$, then we get

$$\begin{aligned}&\int _a^bf_1(x)f_2(x)_{k}J_{a}^{\alpha }f(x)dx \\\le & {} C\left( \int _a^bf_1^p(x)_{k}J_{a}^{\alpha }f(x)dx\right) ^\frac{1}{p} \left( \int _a^bf_2^q(x)_{k}J_{a}^{\alpha }f(x)dx\right) ^\frac{1}{q}, \nonumber \end{aligned}$$

(33)

where

$$\begin{aligned}&C=\sup _{t\in [a,b]}\Bigg \{ \left( \int _a^b \left( x-t\right) ^{\frac{\alpha }{k}-1}f_1(x)f_2(x)dx\right) \\&\left( \int _a^b \left( x-t\right) ^{\frac{\alpha }{k}-1}f_1^p(x)dx\right) ^\frac{-1}{p} \left( \int _a^b \left( x-t\right) ^{\frac{\alpha }{k}-1} f_2^q(x)dx\right) ^\frac{-1}{q}\Bigg \}.\nonumber \end{aligned}$$

(34)

Remark 2.

In Corollary 3, $\alpha =k=1$, Corollary 3 reduces to Corollary 3.2 in [5].

Theorem 6.

Let (a, b) be a finite interval of the real line $\mathbb {R}$ and $\mathfrak {R}(\alpha )> 0$. Let h(x) be an increasing and positive monotone function on (a, b], having a continuous derivative $h'(x)$ on (a, b). Also, let p, q be two real numbers such that $\frac{1}{p}+\frac{1}{q}=1$, $p>1$ and let $f\$be continuous on $[a,b],\ k>0$ and $s\in \mathbb {R}{\setminus }\{-1\}$ . Then

$$\begin{aligned}&\int _a^bf_1(x)f_2(x)\left( _{k}J_{a^+,h}^{\alpha }f\right) (x)dx \\\le & {} C\left( \int _a^bf_3(x)\left( _{k}J_{a^+,h}^{\alpha }f\right) (x)dx\right) ^\frac{1}{p} \left( \int _a^bf_4(x)\left( _{k}J_{a^+,h}^{\alpha }f\right) (x)dx\right) ^\frac{1}{q}, \nonumber \end{aligned}$$

(35)

where

$$\begin{aligned}&C=\sup _{t\in [a,b]}\Bigg \{ \left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)f_1(x)f_2(x)dx\right) \nonumber \\\times & {} \left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)f_3(x)dx\right) ^\frac{-1}{p} \nonumber \\&\left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1} h'(t)f_4(x)dx\right) ^\frac{-1}{q}\Bigg \}. \end{aligned}$$

(36)

Proof.

Applying Theorem 1 with $\varOmega _1=\varOmega _2=(a,b)$, $d\mu _1(t)=dt, d\mu _2(x)=dx$ and the kernel

$$ k(x,t)={\left\{ \begin{array}{ll} \frac{\left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)}{k\varGamma _{k}(\alpha )} &{} \text { if } a\le t\le x \\ 0 &{} \text { if } x<t\le b, \end{array}\right. }$$

then g(x) becomes $\left( _{k}J_{a^+,h}^{\alpha }f\right) (x)$ and so we get desired inequality (35). This completes the proof of Theorem 6.

Corollary 4.

In Theorem 6, setting $f_3(x)=f_1^p(x)$ and $f_4(x)=f_2^q(x)$, we get

$$\begin{aligned}&\int _a^bf_1(x)f_2(x)\left( _{k}J_{a^+,h}^{\alpha }f\right) (x)dx \\\le & {} C\left( \int _a^bf_1^p(x)\left( _{k}J_{a^+,h}^{\alpha }f\right) (x)dx\right) ^\frac{1}{p} \left( \int _a^bf_2^q(x)\left( _{k}J_{a^+,h}^{\alpha }f\right) (x)dx\right) ^\frac{1}{q}, \nonumber \end{aligned}$$

(37)

where

$$\begin{aligned}&C=\sup _{t\in [a,b]}\Bigg \{ \left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)f_1(x)f_2(x)dx\right) \nonumber \\\times & {} \left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)f_1^p(x)dx\right) ^\frac{-1}{p} \nonumber \\&\left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1} h'(t)f_2^q(x)dx\right) ^\frac{-1}{q}\Bigg \}. \end{aligned}$$

(38)

Theorem 7.

Under the assumptions of Theorem 6, we have

$$\begin{aligned}&\int _a^bf_1(x)f_2(x)\left( _{k}J_{b^-,h}^{\alpha }f\right) (x)dx \\\le & {} C\left( \int _a^bf_3(x)\left( _{k}J_{b^-,h}^{\alpha }f\right) (x)dx\right) ^\frac{1}{p} \left( \int _a^bf_4(x)\left( _{k}J_{b^-,h}^{\alpha }f\right) (x)dx\right) ^\frac{1}{q}, \nonumber \end{aligned}$$

(39)

where

$$\begin{aligned}&C=\sup _{t\in [a,b]}\Bigg \{ \left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)f_1(x)f_2(x)dx\right) \nonumber \\\times & {} \left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)f_3(x)dx\right) ^\frac{-1}{p} \nonumber \\&\left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1} h'(t)f_4(x)dx\right) ^\frac{-1}{q}\Bigg \}. \end{aligned}$$

(40)

Proof.

In contrast to Theorem 6, if we take the kernel

$$ k(x,t)={\left\{ \begin{array}{ll} \frac{\left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)}{k\varGamma _{k}(\alpha )} &{} \text { if } x\le t\le b \\ 0 &{} \text { if } a<t\le x, \end{array}\right. }$$

we obtain desired inequality.

Corollary 5.

In Theorem 7, setting $f_3(x)=f_1^p(x)$ and $f_4(x)=f_2^q(x)$, we get

$$\begin{aligned}&\int _a^bf_1(x)f_2(x)\left( _{k}J_{b^-,h}^{\alpha }f\right) (x)dx \\\le & {} C\left( \int _a^bf_1^p(x)\left( _{k}J_{b^-,h}^{\alpha }f\right) (x)dx\right) ^\frac{1}{p} \left( \int _a^bf_2^q(x)\left( _{k}J_{b^-,h}^{\alpha }f\right) (x)dx\right) ^\frac{1}{q}, \nonumber \end{aligned}$$

(41)

where

$$\begin{aligned}&C=\sup _{t\in [a,b]}\Bigg \{ \left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)f_1(x)f_2(x)dx\right) \nonumber \\\times & {} \left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1}h'(t)f_1^p(x)dx\right) ^\frac{-1}{p} \nonumber \\&\left( \int _a^b \left( h(x)-h(t)\right) ^{\frac{\alpha }{k}-1} h'(t)f_2^q(x)dx\right) ^\frac{-1}{q}\Bigg \}. \end{aligned}$$

(42)