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.
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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
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
where
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.:
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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.:
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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.:
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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.:
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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 \(t>0,\ k>0,\ \alpha >0,s\in \mathbb {R}-\{-1\},\)
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
then the inequality
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
then we have
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
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
In equation (19), multiplying both sides by \(\left( _{k}^{s}J_{a}^{\alpha }\left| g^{q}(t) \right| \right) ^{p-1}\), we obtain
Inequality (20) implies inequality
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
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
which is equivalent to
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
which is equivalent to
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
where
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
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
where
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
where
Corollary 3.
In Corollary 2, if we take \(s=0\), then we get
where
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
where
Proof.
Applying Theorem 1 with \(\varOmega _1=\varOmega _2=(a,b)\), \(d\mu _1(t)=dt, d\mu _2(x)=dx\) and the kernel
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
where
Theorem 7.
Under the assumptions of Theorem 6, we have
where
Proof.
In contrast to Theorem 6, if we take the kernel
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
where
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Tomar, M., Agarwal, P., Jain, S., Milovanović, G.V. (2017). Some Reverse Hölder Type Inequalities Involving \((k,s)-\)Riemann-Liouville Fractional Integrals. In: Kalmenov, T., Nursultanov, E., Ruzhansky, M., Sadybekov, M. (eds) Functional Analysis in Interdisciplinary Applications. FAIA 2017. Springer Proceedings in Mathematics & Statistics, vol 216. Springer, Cham. https://doi.org/10.1007/978-3-319-67053-9_29
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