# Fractional Integral Inequalities concerning Extended Bessel Function in the Kernel

Authors: Arshad Hussain; Gauhar Rahman; Jihad Younis; Muhammad Samraiz; Muhammad Iqbal
Journal: Journal of Mathematics
 (2021)

---

## Abstract

The major purpose of this paper is to use the fractional integral operator in terms of extended generalized Bessel function to estimate new fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions. We prove a set of inequalities for the fractional integral operator in terms of extended generalized Bessel function integrals with one and two parameters. Also, we discussed some special cases of the obtained result.


---

## Body

## 1. Introduction

Integral inequalities are particularly essential in the subject of fractional differential equations. In the last few decades, this field has gained more recognition. The important developments in this field have been investigated by various researchers (for example, [1–3]). In [4, 5], the authors presented Grüss-type inequalities and some new inequalities by utilizing Riemann–Liouville (RL) fractional integrals. In [6], certain inequalities for the generalized k,ρ-fractional integral operator were proposed. The generalized Hermite–Hadamard-type inequalities via fractional integral operators were proposed in [7]. By using a family of n positive functions, Dahmani [8] developed several fractional integral inequalities. For the application of fractional integral inequalities, we suggest [9–15]. In recent decades, researchers have devoted their time and effort to the topic of fractional calculus due to its wide applications in a variety of sciences. In [16–21], some numerous developments of fractional integral operators and their applications in various domains can be found. As a result of these novel fractional integral operators, future research will focus on developing new ideas for connecting fractional operators by addressing new fractional integral inequalities. The interested readers are referred to [22–24]. Recently, Kashuri and Liko [25] and Luo et al. [26] presented certain remarkable integral inequalities for the generalized fractional integrals.Ngo et al. [27] proposed the following inequalities:(1)∫01ℏσ+1ζdζ≥∫01ζσℏζdζ,(2)∫01ℏσ+1ζdζ≥∫01ζℏσζdζ,where σ>0 and the function ℏ is positive and continuous on 0,1 with(3)∫x1ℏtdt≥∫θ11ζdζ,θ1∈0,1.Liu et al. [28] proposed the following inequalities:(4)∫θ1θ2ℏσ+κζdζ≥∫θ1θ2ζ−aσℏκζdζ,where σ>0,κ>0, and the function ℏ is positive and continuous on 0,1 with(5)∫θ1ϱℏδζdζ≥∫θ1θ2ζ−aδdζ,ϱ=min1,κ,ζ∈θ1,θ2.Liu et al. [29] presented the following two theorems for integral inequalities.Theorem 1.
Letℏ1andℏ2 be two positive and continuous functions on θ1,θ2,θ1<θ2withℏ1≤ℏ2onθ1,θ2, such that ℏ1/ℏ2ℏ2≠0 is decreasing and ℏ1 is increasing. Assume that ℱis convex function withℱ0=0. Then, the following inequality holds:(6)∫θ1θ2ℏ1tdt∫θ1θ2ℏ2tdt≥∫θ1θ2ℱℏ1tdt∫θ1θ2ℱℏ2tdt.Theorem 2.
Letℏ1, ℏ2, andℏ3be positive and continuous functions onθ1,θ2,θ1<θ2, withℏ1<ℏ2onθ1,θ2, such that ℏ1/ℏ2ℏ2≠0 is decreasing and ℏ1andℏ3 are increasing. Assume thatℱis convex function withℱ0=0. Then, the following inequality holds:(7)∫θ1θ2ℏ1tdt∫θ1θ2ℏ2tdt≥∫θ1θ2ℱℏ1tℏ3tdt∫θ1θ2ℱℏ2tℏ3tdt.The inequalities in (1), (2), and (4) and their several extensions have been widely studied by the researchers [30, 31]. Furthermore, the research of fractional integral inequality is also of extreme significance.Tilahun et al. [32] proposed an extended form of the fractional integral containing Bessel function in the kernel by(8)Jθ1+:ϑ,qω:ς,δ:cϱ;p=∫θ1ϱϱ−λϑJϑ,qς,δ,cwϱ−λς;pℏ1λdλ,ϱ>θ1,where Jϑ,qς,δ,cz;p is the extended Bessel function, defined by(9)Jϑ,qς,δ,cz;p=∑k=0∞Bpδ+qk,c−δcqk−zmBδ,c−δΓςk+ϑ+1k!,ℜp>0,ℜc>0,ℜδ>0,q∈ℕ,where(10)Bpu,v=∫01hu−11−hv−1e−ph/1−hdh,ℜp>0,ℜu>0,ℜv>0.Remark 1.
If we considerp=0in (8), then we get(11)Jθ1+:ϑ,qω:ς,δ:cϱ=∫θ1ϱϱ−λϑJϑ,qς,δ,cwϱ−λςℏ1λdλ,ϱ>θ1.Remark 2.
If we considerω=0, then (11) reduces to the following operator defined by(12)Jθ1+ςϱ=1Γς∫θ1ϱϱ−λς−1ℏ1λdλ,ϱ>θ1.
In this paper, we present fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions by utilizing integral operator (8).


## 2. Main Results

In this section, we present integral inequalities by employing the fractional integral concerning the extended Bessel function in the kernel.Theorem 3.
Letℏ1andℏ2be two continuous and positive functions onθ1,θ2andℏ1≤ℏ2onθ1,θ2. Assume thatℏ1/ℏ2, ℏ2≠0, is decreasing andℏ1is an increasing function onθ1,θ2. Then, for any convex functionℱwithℱ0=0, the following integral inequality holds:(13)Jθ1+;ϑ,qω,ς,δ,cℏ1ϱ;pJθ1+;ϑ,qω,ς,δ,cℏ2ϱ;p≥Jθ1+;ϑ,qω,ς,δ,cℱℏ1ϱ;pJθ1+;ϑ,qω,ς,δ,cℱℏ2ϱ;p,where ς,ϑ,ω∈Cwithp≥0,ℜp>0,ℜς>0, andℜϑ>0.Proof.
By the hypothesis of the theorem, since the functionℱ is convex such that ℱ0=0, the function ℏ1ϱ/ϱ is increasing. Also, ℏ1 is increasing; therefore, ℱℏ1ϱ/ℏ1ϱ is increasing. Obviously, the function ℏ1ϱ/ℏ2ϱ is decreasing. Therefore, for all λ,θ∈θ1,θ2, we have(14)ℱℏ1λℏ1λ−ℱℏ1θℏ1θℏ1θℏ2θ−ℏ1λℏ2λ≥0.
It can also be written as(15)ℱℏ1λℏ1λℏ1θℏ2θ+ℱℏ1θℏ1θℏ1λℏ2λ−ℱℏ1θℏ1θℏ1θℏ2θ−ℱℏ1λℏ1λℏ1λℏ2λ≥0.
The multiplication of (15) by ℏ2λℏ2θ yields(16)ℱℏ1λℏ1λℏ1θℏ2λ+ℱℏ1θℏ1θℏ1λℏ2θ−ℱℏ1θℏ1θℏ1θℏ2λ−ℱℏ1λℏ1λℏ1λℏ2θ≥0.
Now, first conducting the multiplication of (16) by ϱ−λϑJϑ,qς,δ,cwϱ−λς;p and then integrating the estimated inequality with respect to λ over θ1,ϱ, θ1<ϱ≤θ2, we get(17)∫θ1ϱϱ−λϑJϑ,qς,δ,cwϱ−λς;pℱℏ1λℏ1λℏ1θℏ2λdλ+∫θ1ϱϱ−λϑJϑ,qς,δ,cwϱ−λς;pℱℏ1θℏ1θℏ1λℏ2θdλ−∫θ1ϱϱ−λϑJϑ,qς,δ,cwϱ−λς;pℱℏ1θℏ1θℏ1θℏ2λdλ−∫θ1ϱϱ−λϑJϑ,qς,δ,cwϱ−λς;pℱℏ1λℏ1λℏ1λℏ2θdλ≥0.
In view of (8), we get(18)ℏ1θJθ1+;ϑ,qω;ς,δ;cℱℏ1ϱℏ1ϱℏ2ϱ;p+ℱℏ1θℏ1θℏ2θJθ1+;ϑ,qω,ς,δ;cℏ1ϱ;p−ℱℏ1θℏ1θℏ1θJθ1+;ϑ,qω;ς,δ;cℏ2ϱ;p−ℏ2θJθ1+;ϑ,qω;ς,δ;cℱℏ1ϱℏ1ϱℏ2ϱ;p≥0.
Again, conducting the multiplication of (18) by ϱ−θϑJθ1+;ϑ,qω,ς,δ:cwϱ−θς;p and then integrating the resulting inequality with respect to θ over θ1,ϱ, θ1<ϱ≤θ2, we get(19)Jθ1+,ϑ,qω;ς,δ;cℏ1ϱ;pJθ1+,ς,qω;ς,δ;cℱℏ1ϱℏ1ϱℏ2ϱ;p+Jθ1+;ϑ,qω;ς,δ;cℱℏ1ϱℏ1ϱℏ2ϱ;pJθ1+;ϑ;qω;ς,δ;cℏ1ϱ;p≥Jθ1+;ϑ,qω;ς,δ;cℏ2ϱ;pJθ1+;ϑ,qω,ς,δ;cℱℏ1ϱ;p+Jθ1+;ϑ,qω;ς,δ;cℱℏ1ϱ;pJθ1+,ϑ,qω;ς,δ;cℏ2ϱ;p.
In view of (8), it can be written as(20)Jθ1+;ϑ,qω;ς,δ;cℏ1ϱ;pJθ1+;ϑ,qω;ς,δ;cℏ2ϱ;p≥Jθ1+;ϑ,qω;ς,δ;cℱℏ1ϱ;pJθ1+;ϑ,qω;ς,δ;cℱℏ1ϱ/ℏ1ϱℏ2ϱ;p.
Since by hypothesis,ℏ1≤ℏ2 on θ1,θ2 and ℱϱ/ϱ is an increasing function, for λ∈θ1,ϱ, θ1<ϱ≤θ2, we have(21)ℱℏ1λℏ1λ≤ℱℏ2λℏ2λ.
Thus, multiplying (21) by ϱ−λϑJϑ,qς,δ:cwϱ−λς;p and then integrating the estimated inequality with respect to λ over θ1,ϱ, θ1<ϱ≤θ2, yields(22)∫θ1ϱϱ−λϑJϑ,qς,δ;cwϱ−λς;pℱℏ2λℏ1λℏ2λdλ≤∫θ1ϱϱ−λϑJϑ,qς,δ;cwϱ−λς;pℱℏ2λℏ2λℏ2λdλ,which in view of equation (8) follows that(23)Jθ1+;ϑ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ2ϱ;p≤Jθ1+;ϑ,qω;ς,δℱℏ2ϱ;p.
By combining (20) and (23), we get (13).
By consideringp=0 in Theorem 3, we get the following result for the operator given in (11).Corollary 1.
Letℏ1andℏ2be two continuous and positive functions onθ1,θ2andℏ1≤ℏ2 on θ1,θ2. Assume thatℏ1/ℏ2,ℏ2≠0, is decreasing andℏ1is an increasing function onθ1,θ2. Then, for any convex functionℱwithℱ0=0, the following integral inequality holds:(24)Jθ1+;ϑ,qω,ς,δ,cℏ1ϱJθ1+;ϑ,qω,ς,δ,cℏ2ϱ≥Jθ1+;ϑ,qω,ς,δ,cℱℏ1ϱJθ1+;ϑ,qω,ς,δ,cℱℏ2ϱ,where ς,ϑ,ω∈C with ℜς>0 and ℜϑ>0.Remark 3.
By consideringω=0 in Theorem 3, we get the result of Dahmani [33].Remark 4.
Applying Theorem3 for ω=0, ϑ=1, and ϱ=θ2, we get Theorem 1.Theorem 4.
Letℏ1andℏ2be continuous and positive onθ1,θ2, θ1<θ2, andℏ1≤ℏ2onθ1,θ2. Assume thatℏ1/ℏ2,ℏ2≠0, is decreasing function andℏ1is increasing function onθ1,θ2. Then, for any convex function ℱ with ℱ0=0, the following inequality holds:(25)Jθ1+;ϑ,qω;ς,δ,cℏ1ϱ;pJθ1+;κ,qω;ς,δ,cℱℏ2ϱ;p+Jθ1+;κ,qω;ς,δ,cℏ1ϱ;pJθ1+;ϑ,qω;ς,δ,cℱℏ2ϱ;pJθ1+;ϑ,qω;ς,δ,cℏ2ϱ;pJθ1+;κ,qω;ς,δ,cℱℏ1ϱ;p+Jθ1+;κ,qω;ς,δ,cℏ2ϱ;pJθ1+;ϑ,qω;ς,δ,cℱℏ1ϱ;p≥1,where ς,κ,ϑ,ω∈Cwithp≥0,ℜp>0,ℜς>0,ℜϑ>0, and ℜκ>0.Proof.
By the hypothesis of the theorem,ℱ is convex function and ℱ0=0, and the function ℏ1ϱ/ϱ is increasing. Also, since ℏ1 is increasing, ℱℏ1ϱ/ℏ1ϱ is obviously increasing function. As ℏ1ϱ/ℏ2ϱ is decreasing function, for all θ∈θ1,ϱ, θ1<ϱ≤θ2. Thus, by taking product of (18) and ϱ−θκJκ,qς,δ;cwϱ−θς;p and integrating the resulting inequality with respect to θ over θ1,ϱ, θ1<ϱ≤θ2, we get(26)Jθ1+;ϑ,qω,ς,δ,cℏ1ϱJθ1+;κ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ2ϱ;p+Jθ1+;κ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ2ϱ;pJθ1+;ϑ,qω,ς,δ,cℏ1ϱ;p≥Jθ1+;ϑ,qω;ς,δ:cℏ2ϱJθ1+;κ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ1ϱ;p+Jθ1+;κ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ1ϱ;pJθ1+;ϑ,qω;ς,δ:cℏ2ϱ;p.
Also, on the other hand, the functionℏ1≤ℏ2 on θ1,θ2 and ℱϱ/ϱ is an increasing function. Therefore, for, θ∈θ1,ϱ, θ1<ϱ≤θ2, we have(27)ℱℏ1θℏ1θ≤ℱℏ2θℏ2θ.
The multiplication of (27) by ϱ−θκJϑ,qς;δ,cwϱ−θς;p and integration of the resultant inequality with respect to θ over θ1,ϱ, θ1<ϱ≤θ2, yields(28)∫θ1ϱϱ−θκJκ,qς;δ,cwϱ−θς;pℱℏ1θℏ1θℏ2θdθ≤∫θ1ϱϱ−θκJκ,qς;δ,cwϱ−κς;pℱℏ2θℏ2θℏ2θdθ,which with the aid of (8) follows that(29)Jθ1+;κ,qω,ς,δ,cℱℏ1ϱℏ1ϱℏ2ϱ;p≤Jθ1+;κ,qω;ς,δ:cℱℏ2ϱ;p.
Similarly, one can get(30)Jθ1+;ϑ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ2ϱ;p≤Jθ1+;ϑ,qω;ς,δ,;cℱℏ2ϱ;p.
By combining (26), (29), and (30), we get the required inequality.
By consideringp=0 in Theorem 4, we get the following result for the operator given in (11).Corollary 2.
Letℏ1andℏ2be continuous and positive onθ1,θ2,θ1<θ2, andℏ1≤ℏ2onθ1,θ2. Assume thatℏ1/ℏ2,ℏ2≠0, is decreasing function andℏ1is increasing function onθ1,θ2. Then, for any convex functionℱwithℱ0=0, the following inequality holds:(31)Jθ1+;ϑ,qω;ς,δ,cℏ1ϱJθ1+;κ,qω;ς,δ,cℱℏ2ϱ+Jθ1+;κ,qω;ς,δ,cℏ1ϱJθ1+;ϑ,qω;ς,δ,cℱℏ2ϱJθ1+;ϑ,qω;ς,δ,cℏ2ϱJθ1+;κ,qω;ς,δ,cℱℏ1ϱ+Jθ1+;κ,qω;ς,δ,cℏ2ϱ;pJθ1+;ϑ,qω;ς,δ,cℱℏ1ϱ≥1,where ϑ,κ,ς,ω∈Cwithℜϑ>0,ℜκ>0, andℜς>0.Remark 5.
By consideringω=0 in Theorem 4, we get the result of Dahmani [33].Remark 6.
Applying Theorem4 for ω=0, ϑ=κ=1, and ϱ=θ2, we get Theorem 1.Remark 7.
Applying Theorem4 for ϑ=κ, we get Theorem 3.Theorem 5.
Letℏ1,ℏ2, andℏ3be continuous and positive functions onθ1,θ2,θ1<θ2, andℏ1≤ℏ2onθ1,θ2. If the functionℏ1/ℏ2,ℏ2≠0, is decreasing,ℏ1andℏ3are increasing functions onθ1,θ2. Then, ifℱis any convex function withℱ0=0, the following inequality holds:(32)Jθ1+;ϑ,qω;ς,δ:cℏ1ϱ;pJθ1+;ϑ,qω;ς,δ,cℏ2ϱ;p≥Jθ1+;ϑ,qω,ς,δ:cℱℏ1ϱℏ3ϱ;pJθ1+;ϑ,qω,ς,δ:cℱℏ2ϱℏ3ϱ;p,where ϑ,ς,ω∈C, with p≥0,ℜϑ>0,ℜς>0, and ℜp>0.Proof.
By the hypothesis of the theorem,ℱ is convex function such that ℱ0=0 and the function ℱϱ/ϱ is increasing. Also, since f is increasing, ℱℏ1ϱ/ℏ1ϱ is increasing. It is clear that ℏ1ϱ/ℏ2ϱ is decreasing. Thus, for all λ,θ∈θ1,ϱ, θ1<ϱ≤θ2, we have(33)ℱℏ1λℏ1λℏ3λ−ℱℏ1θℏ1θℏ3θℏ1θℏ2λ−ℏ1λℏ2θ≥0.
It can be written as(34)ℱℏ1λℏ3λℏ1λℏ1θℏ2λ+ℱℏ1θℏ3θℏ1θℏ1λℏ2θ−ℱℏ1θℏ3θℏ1θℏ1θℏ2λ−ℱℏ1λℏ3λℏ1λℏ1λℏ2θ≥0.
The multiplication of (34) by ϱ−λϑJϑ,qς,δ;cwϱ−λς;p and integration of the resultant inequality with respect to λ over θ1,ϱ, θ1<ϱ≤θ2, yields(35)∫θ1ϱϱ−λϑJϑ,qς,δ;cwϱ−λς;pℱℏ1λℏ1λℏ1θℏ2λℏ3λdλ+∫θ1ϱϱ−λϑJϑ,qς,δ:cwϱ−λς;pℱℏ1θℏ1θℏ1λℏ2θℏ3θdλ−∫θ1ϱϱ−λϑJϑ,qς,δ:cwϱ−λς;pℱℏ1θℏ1θℏ1θℏ2λℏ3θdλ−∫θ1ϱϱ−λϑJϑ,qς,δ,cwϱ−λς;pℱℏ1λℏ1λℏ1λℏ2θℏ3λdλ≥0.
With the aid of (8), we can write the above equation as(36)ℏ1θJϑ,qς,δ;cℱℏ1ϱℏ1ϱℏ2ϱhϱ;p+ℱℏ1θℏ1θℏ2θhθJϑ,qς,δ;cℏ1ϱ;p−ℱℏ1θℏ1θℏ1θℏ3θJϑ,qς,δ;cℏ2ϱ;p−ℏ2θJϑ,qς,δ;cℱℏ1ϱℏ1ϱℏ2ϱℏ3ϱ;p≥0.
Again, the multiplication of (36) by ϱ−θϑJϑ,qς,δ;cwϱ−θς;pℏ2θℏ3θ and integration of the estimated inequality with respect to θ over θ1,ϱ, θ1<ϱ≤θ2, and then applying (8) yields(37)Jθ1+;ϑ,qω,ς,δ;cℏ1ϱ;pJϑ,qς,δ;cℱℏ1ϱℏ1ϱℏ2ϱℏ3ϱ;p+Jθ1+;ϑ,qω,ς,δ;cℱℏ1ϱℏ1ϱℏ2ϱℏ3ϱ;pJϑ,qς,δ;cℏ1ϱ;p≤Jθ1+;ϑ,qω,ς,δ;cℏ2ϱ;pJϑ,qς,δ;cℏ1ϱℏ3ϱ;p+Jθ1+;ϑ,qω,ς,δ;cℱℏ1ϱℏ3ϱ;pJϑ,qς,δ;cℏ2ϱ;p.
It can also be written as(38)Jθ1+;ϑ,qω,ς,δ;cℏ1ϱ;pJθ1+;ϑ,qω,ς,δ;cℏ2ϱ;p≥Jθ1+;ϑ,qω,ς,δ;cℱℏ1ϱℏ3ϱJθ1+;ϑ,qω,ς,δ;cℱℏ1ϱ;p/ℏ1ϱℏ2ϱℏ3ϱ;p.
Hence, by using (23) and (38), we get the required assertion.
By consideringp=0 in Theorem 5, we get the following result for the operator given in (11).Corollary 3.
Letℏ1,ℏ2, andℏ3be continuous and positive functions onθ1,θ2,θ1<θ2, andℏ1≤ℏ2onθ1,θ2. If the functionℏ1/ℏ2,ℏ2≠0, is decreasing,ℏ1andℏ3are increasing functions onθ1,θ2. Then, ifℱis any convex function withℱ0=0, the following inequality holds:(39)Jθ1+;ϑ,qω;ς,δ:cℏ1ϱJθ1+;ϑ,qω;ς,δ,cℏ2ϱ≥Jθ1+;ϑ,qω,ς,δ:cℱℏ1ϱℏ3ϱJθ1+;ϑ,qω,ς,δ:cℱℏ2ϱℏ3ϱ,where ϑ,ς,ω∈C, with ℜϑ>0 and ℜς>0.Remark 8.
By consideringω=0 in Theorem 5, we get the result of Dahmani [33].Remark 9.
Applying Theorem5 for ω=0, ϑ=1 and ϱ=θ2, we get Theorem 2.Theorem 6.
Letℏ1,ℏ2, andℏ3be continuous and positive functions onθ1,θ2,θ1<θ2, andℏ1≤ℏ2onθ1,θ2. Ifℏ1/ℏ2is decreasing,ℏ1andℏ3are increasing functions onθ1,θ2. Letℱbe any convex function withℱ0=0. Then, the following inequality holds:(40)Jθ1+ϑ,qω;ς;δ,cℏ1ϱ;pJθ1+,κ,qω;ς,δ:cℱℏ2ϱℏ3ϱ;p+Jθ1+,κ,qω;ς,δ:cℏ1ϱ;pJθ1+,ϑ,qω;ς,δ,cℱℏ2ϱℏ3ϱ;pJθ1+;ϑ,qω;ς,δ:cℏ2ϱ;pJθ1+,κ,qω;ς,δ:cℱℏ1ϱℏ3ϱ;p+Jθ1+;κ,qω,ς,δ:cℏ2ϱ;pJθ1+;ϑ,qω;ς,δ:cℱℏ1ϱℏ3ϱ;p≥1,where κ,ϑ,ς,ω∈Cwithp≥0,ℜκ>0,ℜp>0,ℜς>0, andℜϑ>0.Proof.
Taking multiplication of (36) by ϱ−θκJκ,qς;δ;cwϱ−θς;p and integrating the inequality with respect to θ over θ1,ϱ, θ1<ϱ≤θ2, we obtain(41)Jθ1+;ϑ,qω;ς,δ:cℏ1ϱ;pJθ1+;κ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ2ϱℏ3ϱ;p+Jθ1+;κ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ2ϱℏ3ϱ;pJθ1+;ϑ,qω;ς,δ:cℏ1ϱ;p≤Jθ1+;ϑ,qω;ς,δ:cℏ2ϱ;pJθ1+;κ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ2ϱℏ3ϱ;p+Jθ1+;κ,qω;ς,δ:cℱℏ1ϱℏ1ϱℏ1ϱℏ3ϱ;pJθ1+;ϑ,qω;ς,δ:cℏ2ϱ;p.
Hence, by using (29), (30), and (41), we get the desired assertion.
By consideringp=0 in Theorem 6, we get the following result for the operator given in (11).Corollary 4.
Letℏ1,ℏ2, andℏ3be continuous and positive functions onθ1,θ2,θ1<θ2, andℏ1≤ℏ2onθ1,θ2. Ifℏ1/ℏ2is decreasing,ℏ1andℏ3are increasing functions onθ1,θ2. Letℱbe any convex function withℱ0=0. Then, the following inequality holds:(42)Jθ1+ϑ,qω;ς;δ,cℏ1ϱJθ1+,κ,qω;ς,δ:cℱℏ2ϱℏ3ϱ+Jθ1+,κ,qω;ς,δ:cℏ1ϱJθ1+,ϑ,qω;ς,δ,cℱℏ2ϱℏ3ϱJθ1+;ϑ,qω;ς,δ:cℏ2ϱJθ1+,κ,qω;ς,δ:cℱℏ1ϱℏ3ϱ+Jθ1+;κ,qω,ς,δ:cℏ2ϱ;pJθ1+;ϑ,qω;ς,δ:cℱℏ1ϱℏ3ϱ≥1,where κ,ϑ,ς,ω∈C with ℜκ>0,ℜϑ>0, and ℜϑ>0.Remark 10.
By consideringω=0 in Theorem 6, we get the result of Dahmani [33].Remark 11.
Applying Theorem6 for ω=0, ϑ=κ=1, and ϱ=θ2, we get Theorem 2.Remark 12.
Applying Theorem6 for ϑ=κ, we get Theorem 5.


## 3. Concluding Remarks

In this present investigation, we have investigated some new generalized integral inequalities for the operator involving generalized extended Bessel function in the kernel. The integral inequalities presented in this paper are more general than the existing inequalities given in the literature. The special cases of the obtained result can be found in the work of [29, 33]. One can obtain various types of integral inequalities by employing integral operator (8).



---

*Source: 7325102-2021-10-20.xml*