# Chebyshev-Type Inequalities Involving (k,ψ)-Proportional Fractional Integral Operators

Authors: Bhagwat R. Yewale; Deepak B. Pachpatte; Tariq A. Aljaaidi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Expanding the analytical aspect of mathematics enables researchers to study more cosmic phenomena, especially with regard to the applied sciences related to fractional calculus. In the present paper, we establish some Chebyshev-type inequalities in the case synchronous functions. In order to achieve our goals, we usek,ψ-proportional fractional integral operators. Moreover, we present some special cases.


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## Body

## 1. Introduction

The generalization of differentiation and integration of integer order to noninteger order is the subject of study in Fractional Calculus (FC). The theory of FC has become a subject of great interest with many applications, such as viscoelasticity [1], bioengineering [2], and control theory [3]. There are several definitions of fractional operators appearing in the literature, ranging from the most well-known Riemann-Liouville and Caputo to others such as Hadamard, Erdèlyi-Kober, Riesz, and Prabhakar [4, 5]. Miller and Ross [6] and Samko et al. [7] have written a research monograph that goes into great detail about fractional operators, including their properties and applications.In the last few decades, the development of FC theory has been taken into consideration by many researchers. Since then, researchers have primarily focused on the generalizations of fractional operators by exhibiting different kernels, which has made a significant contribution to the development of FC. In particular, Jarad et al. [8] has contributed significantly to the study of FC by introducing generalized proportional fractional operators through exponential kernel, while in [9], the authors extended this work and defined the proportional fractional operator of a function with respect to the another function. Furthermore, in [10], Aljaaidi et al. extended the studies on proportional fractional operators and proposed new fractional operators known as k,ψ-proportional fractional operators.On the other hand, due to the importance of fractional operators, many different types of inequalities have been established by researchers using various fractional operators. For instance, Set et al. [11] established Pólya-Szegö-type inequalities by means of a generalized proportional Hadamard fractional operator. Rashid et al. [12] presented reverse Minkowski’s type inequalities by using generalized proportional fractional operators. In [10], authors studied Pólya-Szegö-type inequalities and Grüss-type inequalities with the help of k,ψ-proportional fractional operators. In recent years, many researchers have been working in the direction of estimating the fractional version of various inequalities [13–19] and the references given there.In 1882, Chebyshev introduced the following functional (see [20]):Letf and g be an integrable function on the interval a,b. Then,
(1)Wf,g=1b−a∫abfxgxdx−1b−a∫abfxdx1b−a∫abgxdx.Iff and g are synchronous, that is,
(2)fx−fygx−gy≥0,forx,y∈a,b,then(3)Wf,g≥0.In [20], authors introduced the following weighted version of the Chebyshev functional which is the extension of (1):Letf and g be the integrable function on a,b and p be the positive and integrable function on a,b. Then,
(4)Wf,g,p=∫abpxdx∫abpxfxgxdx−∫abpxfxdx∫abpxgxdx.Many researchers have given considerable attention to both functionals, and a number of inequalities and a number of extensions, generalizations, and variants have appeared in the literature over the past few decades (see [21–26]).Furthermore, takingq as a positive and integrable function on a,b, the extended Chebyshev functional is given as follows (see [27]):
(5)Wf,g,p,q=∫abqxdx∫abpxfxgxdx+∫abpxdx∫abqxfxgxdx−∫abpxfxdx∫abqxgxdx−∫abqxfxdx∫abpxgxdx.In the literature, many authors provide the results about the Chebyshev-type inequality for various fractional operators. In 2009, Belarbi and Dahmani in [28] presented Chebyshev-type inequalities related to functional (1) by employing the Riemann-Liouville definition. We also note that the Chebyshev-type inequalities for the extended Chebyshev functional established by Dahmani in [29] used the same operator. In 2016, Sarikaya et al. in [30] introduced a new definition k,s-Riemann-Liouville fractional and obtained Chebyshev-type inequalities for it in the case of functional (1). Chebyshev-type inequalities were studied by Set et al. in [31] via extended generalized fractional operators. In [32], authors used the generalized Katugampola operators to establish integral inequalities for Chebyshev and extended Chebyshev functionals. In the literature, a number of mathematicians have devoted their efforts to study Chebyshev-type inequalities using various fractional operators [33–38].Motivated by the above works, our aim in this paper is to establish some new Chebyshev-type inequalities. We usek,ψ-proportional fractional operators to establish our main results. The rest of the paper is organized as follows: In Section 2, we give some preliminaries and definitions which will be useful in the sequel. In Section 3, we prove some Chebyshev-type inequalities related to functional (1) and the extended Chebyshev functional. In Section 4, we give the concluding remark.


## 2. Preliminaries

In this section, we give some basic definitions, preliminaries which are useful for our subsequent discussions:Definition 1 (see [8]).
Letα>0,ν∈0,1. Let f be an integrable function on a,b. Then, generalized proportional fractional integral of f of order α is given by
(6)Iaα,νft=1ναΓα∫ateν−1/νt−ωt−ωα−1fωdω,
whereΓα=∫0∞e−μμα−1dμ,μ>0 is the Gamma function.Definition 2 (see [9]).
Letα>0,ν∈0,1. Let f be an integrable function and ψ be a continuous increasing function on a,b. Then, generalized proportional fractional integral of f with respect to ψ of order α is given by
(7)ψIaα,νft=1ναΓα∫ateν−1/νψt−ψωψt−ψωα−1ψ′ωfωdω,
whereΓα=∫0∞e−μμα−1dμ, μ>0 is the Gamma function.Definition 3 (see [10]).
Letα>0,k>0,ν∈0,1. Let f be an integrable function and ψ be a continuous increasing function on a,b. Then, the k,ψ-proportional fractional integral of f with respect to ψ of order α is given by
(8)k,ψIaα,νft=1να/kkΓkα∫ateν−1/νψt−ψωψt−ψωα/k−1ψ′ωfωdω,
whereΓkα=∫0∞e−μk/kμα−1dμ, μ>0, is the k-Gamma function.


## 3. Chebyshev-Type Inequalities

This section deals with the Chebyshev-type integral inequalities for synchronous functions.Theorem 4.
Letα>0. Assume that f and g are integrable and synchronous functions on 0,∞. Then
(9)k,ψI0α,νfgt≥k,ψI0α,νftk,ψI0α,νgtk,ψI0α,ν1.Proof.
Sincef and g are synchronous functions on 0,∞, for all ω,μ≥0, we have
(10)fω−fμgω−gμ≥0.
Then,(11)fωgω+fμgμ≥fωgμ+fμgω.
Now, on both sides of (11), taking multiplication by eν−1/νψt−ψωψ′ω/να/kkΓkαψt−ψω1−α/k, where ω∈0,t; then, integrating with respect to ω from 0 to t, we obtain
(12)1να/kkΓkα∫0teν−1/νψt−ψωψ′ωψt−ψω1−α/kfωgωdω+fμgμνα/kkΓkα∫0teν−1/νψt−ψωψ′ωψt−ψω1−α/kdω≥gμνα/kkΓkα∫0teν−1/νψt−ψωψ′ωψt−ψω1−α/kfωdω+fμνα/kkΓkα∫0teν−1/νψt−ψωψ′ωψt−ψω1−α/kgωdω.
Consequently,(13)Ik,ψ0α,νfgt+fμgμIk,ψ0α,ν1≥gμIk,ψ0α,νft+fμIk,ψ0α,νgt.
Multiplying both the sides of (13) by eν−1/νψt−ψμψ′μ/να/kkΓkαψt−ψμ1−α/k, where μ∈0,t; then, integrating with respect to μ from 0 to t, we get
(14)Ik,ψ0α,νfgtIk,ψ0α,ν1+k,ψI0α,νfgtIk,ψ0α,ν1≥Ik,ψ0α,νftIk,ψ0α,νgt+Ik,ψ0α,νftIk,ψ0α,νgt.
Hence,(15)k,ψI0α,νfgt≥k,ψI0α,νftk,ψI0α,νgtk,ψI0α,ν1.Theorem 5.
Letα>0 and β>0. Assume that f and g are integrable and synchronous functions on 0,∞. Then,
(16)Ik,ψ0α,ν1Ik,ψ0β,νfgt+Ik,ψ0β,ν1Ik,ψ0α,νfgt≥Ik,ψ0α,νftIk,ψ0β,νgt+Ik,ψ0β,νftIk,ψ0α,νgt.Proof.
Sincef and g are synchronous on 0,∞, using the same argument as in Theorem (9), we obtain
(17)Ik,ψ0α,νfgt+fμgμIk,ψ0α,ν1≥gμIk,ψ0α,νft+fμIk,ψ0α,νgt.
Now, on both sides of (17), taking multiplication by eν−1/νψt−ψμψ′μ/νβ/kkΓkβψt−ψμ1−β/k, where μ∈0,t; then, integrating with respect to μ from 0 to t, we have
(18)Ik,ψ0α,νfgt1νβ/kkΓkβ∫0teν−1/νψt−ψμψ′μψt−ψμ1−β/kdμ+I0α,ν11νβ/kkΓkβ∫0teν−1/νψt−ψμψ′μψt−ψμ1−β/kfμgμdμ≥Ik,ψ0α,νft1νβ/kkΓkβ∫0teν−1/νψt−ψμψ′μψt−ψμ1−β/kgμdμ+Ik,ψ0α,νgt1νβ/kkΓkβ∫0teν−1/νψt−ψμψ′μψt−ψμ1−β/kfμdμ.
From above,(19)Ik,ψ0α,ν1Ik,ψ0α,νfgt+Ik,ψ0α,ν1Ik,ψ0α,νfgt≥Ik,ψ0α,νftIk,ψ0β,νgt+Ik,ψ0β,νftIk,ψ0α,νgt.
In the next result, we prove some integral inequalities for synchronous function related to the extended Chebyshev functional in the case of one fractional parameters. First, we prove the following lemma in order to use it to prove our next theorem.Lemma 6.
Letα>0. Assume that f and g be integrable and synchronous functions on 0,∞. Suppose ζ, η: 0,∞⟶0,∞. Then,
(20)Ik,ψ0α,νζfgtIk,ψ0α,νηt+Ik,ψ0α,νζtIk,ψ0α,νηfgt≥Ik,ψ0α,νζftIk,ψ0α,νηgt+Ik,ψ0α,νζgtIk,ψ0α,νηft.Proof.
Sincef and g are synchronous functions on 0,∞; for all ω,μ≥0, we have
(21)fω−fμgω−gμ≥0.
Then,(22)fωgω+fμgμ≥fωgμ+fμgω.
On both sides of (22), taking multiplication by eν−1/νψt−ψωψ′ωζω/να/kkΓkαψt−ψω1−α/k, where ω∈0,t; then, integrating with respect to ω from 0 to t, we get
(23)Ik,ψ0α,νζfgt+fμgμIk,ψ0α,νζt≥gμIk,ψ0α,νfζt+fμIk,ψ0α,νgζt.
Multiplying both sides of (23) by eν−1/νψt−ψμψ′μημ/να/kkΓkαψt−ψμ1−α/k, where μ∈0,t; then, integrating with respect to μ from 0 to t, we get
(24)Ik,ψ0α,νζfgtIk,ψ0α,νηt+Ik,ψ0α,νηfgtIk,ψ0α,νζt≥Ik,ψ0α,νηgtIk,ψ0α,νfζt+Ik,ψ0α,νηftIk,ψ0α,νζgt.Theorem 7.
Letα>0. Assume that f and g are integrable and synchronous functions on 0,∞. Suppose p,q,r: 0,∞⟶0,∞. Then, for t≥0,
(25)2Ik,ψ0α,νrtIk,ψ0α,νptIk,ψ0α,νqfgt+Ik,ψ0α,νqtIk,ψ0α,νpfgt+2k,ψI0α,νptIk,ψ0α,νrfgtIk,ψ0α,νqt≥Ik,ψ0α,νrtIk,ψ0α,νpftIk,ψ0α,νqgt+Ik,ψ0α,νpgtIk,ψ0α,νqft+Ik,ψ0α,νptIk,ψ0α,νrftIk,ψ0α,νqgt+Ik,ψ0α,νrgtIk,ψ0α,νqft+Ik,ψ0α,νqtIk,ψ0α,νrftIk,ψ0α,νpgt+Ik,ψ0α,νrgtIk,ψ0α,νpft.Proof.
Substitutingζ=p,η=q in (20), then multiplying by Ik,ψ0α,νrt, we get
(26)Ik,ψ0α,νrtIk,ψ0α,νptIk,ψ0α,νqfgt+Ik,ψ0α,νrtIk,ψ0α,νqtIk,ψ0α,νpfgt≥Ik,ψ0α,νrtIk,ψ0α,νpftIk,ψ0α,νqgt+Ik,ψ0α,νrtIk,ψ0α,νpgtIk,ψ0α,νqft.
Puttingζ=r,η=q in (20), then multiplying by Ik,ψ0α,νpt, we obtain
(27)Ik,ψ0α,νptIk,ψ0α,νrtIk,ψ0α,νqfgt+Ik,ψ0α,νptIk,ψ0α,νqtIk,ψ0α,νrfgt≥Ik,ψ0α,νptIk,ψ0α,νrftIk,ψ0α,νqgt+Ik,ψ0α,νptIk,ψ0α,νrgtIk,ψ0α,νqft.
Now puttingζ=r,η=p in (20), then multiplying by Ik,ψ0α,νqt, we have
(28)Ik,ψ0α,νqtIk,ψ0α,νrtIk,ψ0α,νpfg+Ik,ψ0α,νqtIk,ψ0α,νptIk,ψ0α,νrfg≥Ik,ψ0α,νqtIk,ψ0α,νrftIk,ψ0α,νpgt+Ik,ψ0α,νqtIk,ψ0α,νrgtIk,ψ0α,νpft.
Adding (26), (27), and (28), we get (25).
In the next result, we prove some integral inequalities for the synchronous function related to the extended Chebyshev functional using two fractional parameters. First, we prove the following lemma in order to use it to prove our next theorem.Lemma 8.
Letα>0,β>0. Assume that f and g are integrable and synchronous functions on 0,∞ and ζ, η: 0,∞⟶0,∞. Then,
(29)Ik,ψ0α,νζtIk,ψ0β,νηfgt+Ik,ψ0β,νηtIk,ψ0α,νζfgt≥Ik,ψ0α,νζftIk,ψ0β,νηgt+Ik,ψ0α,νζgtIk,ψ0β,νηft.Proof.
Sincef and g are synchronous on 0,∞, by using same arguments as in the proof of Lemma (9), we have
(30)Ik,ψ0α,νζfgt+fμgμIk,ψ0α,νζt≥gμIk,ψ0α,νfζt+fμIk,ψ0α,νζgt.
Multiplying both sides of (30) by eν−1/νψt−ψμψ′μημ/νβ/kkΓkβψt−ψμ1−β/k, where μ∈0,t; then, integrating with respect to μ from 0 to t, we get
(31)Ik,ψ0α,νζfgtIk,ψ0β,νηt+Ik,ψ0β,νηfgtIk,ψ0α,νζt≥Ik,ψ0β,νηgtIk,ψ0α,νfζt+Ik,ψ0β,νηftIk,ψ0α,νζgt.Theorem 9.
Letα>0. Assume that f and g are integrable and synchronous functions on 0,∞. Suppose p,q,r: 0,∞⟶0,∞. Then,
(32)Ik,ψ0α,νrtIk,ψ0α,νqtIk,ψ0β,νpfgt+2Ik,ψ0α,νptIk,ψ0β,νqfgt+Ik,ψ0β,νqtIk,ψ0β,νpfgt+Ik,ψ0β,νrfgtIk,ψ0α,νptIk,ψ0β,νqt+Ik,ψ0β,νptIk,ψ0α,νqt≥Ik,ψ0α,νrtIk,ψ0α,νpftIk,ψ0β,νqgt+Ik,ψ0β,νqftIk,ψ0α,νpgt+Ik,ψ0α,νpt+Ik,ψ0α,νrftIk,ψ0β,νqgt+Ik,ψ0β,νqftIk,ψ0α,νrgt+Ik,ψ0α,νqtIk,ψ0α,νrftIk,ψ0β,νpgt+Ik,ψ0β,νpftIk,ψ0α,νrgt.Proof.
Substitutingζ=p,η=q in (29), then multiplying both sides by Ik,ψ0α,νrt, we get
(33)Ik,ψ0α,νrtIk,ψ0α,νptIk,ψ0β,νqfgtIk,ψ0α,νrtIk,ψ0β,νqtIk,ψ0α,νpfgt≥Ik,ψ0α,νrtIk,ψ0α,νpftIk,ψ0β,νqgt+Ik,ψ0α,νrtIk,ψ0β,νpgtIk,ψ0α,νqft.
Puttingζ=r,η=q in (29), then multiplying by Ik,ψ0α,νpt, we obtain
(34)Ik,ψ0α,νptIk,ψ0α,νrtIk,ψ0β,νqfgt+Ik,ψ0α,νptIk,ψ0β,νqtIk,ψ0α,νrfgt≥Ik,ψ0α,νptIk,ψ0α,νrftIk,ψ0β,νqgt+Ik,ψ0α,νptIk,ψ0β,νqftIk,ψ0α,νrgt.
Now puttingζ=r,η=p in (29), then multiplying by Ik,ψ0α,νqt, we have
(35)Ik,ψ0α,νqtIk,ψ0α,νrtIk,ψ0β,νpfgt+Ik,ψ0α,νqtIk,ψ0β,νptIk,ψ0α,νrfgt≥Ik,ψ0α,νqtIk,ψ0α,νrftIk,ψ0β,νpgt+Ik,ψ0α,νqtIk,ψ0β,νpftIk,ψ0α,νqgt.
Adding (33), (34), and (35), we get (32).Remark 10.
If we putα=β, then inequality (32) reduces to the inequality (25).


## 4. Concluding Remark

In this paper, we obtained some Chebyshev-type inequalities for synchronous functions by employingk,ψ-proportional fractional operators. The main objective of this paper is to contribute to a greater extent of studies within the theory of mathematical inequalities. The outcomes of this paper are generalized in nature. The k,ψ-proportional fractional operators used in the present paper reduces to the various existing fractional integral operators. If we put
(i)
k=1 and ψt=t; then, k,ψ-proportional fractional operators reduce to the fractional proportional operators [8](ii)
k=1; then, k,ψ-proportional fractional operators reduce to the ψ-fractional proportional operators [9](iii)
k=1 and ν=1; then, k,ψ-proportional fractional operators reduce to the ψ-Riemann-Liouville fractional operators [4](iv)
ν=1 and ψt=t; then, k,ψ-proportional fractional operators reduce to the k-Riemann-Liouville fractional operators [39](v)
k=1, ν=1, and ψt=t; then, k,ψ-proportional fractional operators reduce to the well-known Riemann-Liouville fractional operators [4].



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