# Boundedness of Fractional Integral Operators Containing Mittag-Leffler Function via Exponentiallys-Convex Functions

Authors: Gang Hong; G. Farid; Waqas Nazeer; S. B. Akbar; J. Pečarić; Junzhong Zou; Shengtao Geng
Journal: Journal of Mathematics
 (2020)

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## Abstract

The main objective of this paper is to obtain the fractional integral operator inequalities which provide bounds of the sum of these operators at an arbitrary point. These inequalities are derived fors-exponentially convex functions. Furthermore, a Hadamard inequality is obtained for fractional integrals by using exponentially symmetric functions. The results of this paper contain several such consequences for known fractional integrals and functions which are convex, exponentially convex, and s-convex.


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## Body

## 1. Introduction

Convex functions play an important role in many areas of mathematics. They are important especially in the study of optimization problems, theory of inequalities, mathematical analysis, statistical analysis, operation research, and so on. The analytical definition of convex function motivated the authors to define more such functions theoretically; for example, the terms quasi-convex,m-convex, s-convex, h-convex, α,m-convex, and h−m-convex functions [1] are defined by extending or generalizing inequality (1). For this paper, we will use exponentially s-convex functions which include exponentially convex, s-convex, and convex functions.Definition 1.
A functionf:K⊆ℝ⟶ℝ, where K is an interval in ℝ, is said to be convex function if the following inequality holds:(1)fta+1−tb≤tfa+1−tfb,for all a,b∈K and t∈0,1.Definition 2 (see [2]).
A functionf:K⊆ℝ⟶ℝ, where K is an interval in ℝ, is said to be exponentially convex function if(2)fta+1−tb≤tfaeαa+1−tfbeαbholds for all a,b∈K, t∈0,1 and α∈ℝ. If the inequality in (2) is reversed, then f is called exponentially concave.
A generalization of convex function defined on the right half of the real line is thes-convex function defined as follows:Definition 3.
(see [3]). Let s∈0,1. A function f:0,∞⟶ℝ is said to be s-convex function in the second sense if(3)fta+1−tb≤tsfa+1−tsfbholds for all a,b∈0,∞ and t∈0,1.
It is noted that I-convex function is convex. A further generalization of the above defined functions is given as follows:Definition 4.
(see [4]). Let s∈0,1 and K⊆0,∞ be an interval. A function f:K⟶ℝ is said to be exponentially s-convex in the second sense if(4)fta+1−tb≤tsfaeαa+1−tsfbeαbholds for all a,b∈K, t∈0,1 and α∈ℝ. If the inequality in (4) is reversed, then f is called exponentially s-concave.
For utilizations of exponentially convex functions, one can see [2, 4–7]. Our aim in this paper is to utilize exponentially s-convex functions for establishing bounds of fractional integral operators with kernel Mittag-Leffler function. The Mittag-Leffler function denoted by Eσt was introduced by Mittag-Leffler [8] in 1903:(5)Eσt=∑n=0∞tnΓσn+1,where σ,t∈ℂ, Γ⋅ is the gamma function, and ℜσ≥0.
The Mittag-Leffler function is a direct generalization of the exponential function to which it reduces forσ=1. In the solution of fractional integral and fractional differential equations, it arises naturally. Due to its importance and utilizations, Mittag-Leffler function has been generalized by many authors. By direct involvement in the problems of physics, chemistry, biology, engineering, and other applied sciences, Mittag-Leffler function and its generalizations have successful applications. Recently, in [9], Andrić et al. introduced an extended generalized Mittag-Leffler function which is defined as follows:Definition 5.
Letμ,σ,l,γ,c∈ℂ, ℜμ,ℜσ,ℜl>0, and ℜc>ℜγ>0 with p≥0, δ>0, and 0<k≤δ+ℜμ. Then, the extended generalized Mittag-Leffler function is defined by(6)Eμ,σ,lγ,δ,k,ct;p=∑n=0∞βpγ+nk,c−γβγ,c−γcnkΓμn+σtnlnδ,where βp is the generalized beta function defined as follows:(7)βpx,y=∫01tx−11−ty−1e−p/t1−tdt,and cnk is the Pochhammer symbol defined by cnk=Γc+nk/Γc.
From (4), one can obtain the Mittag-Leffler functions defined in [10–14], see Remark 1.3 of [32].Lemma 1.
(see [9]). If m∈ℕ and ω,μ,σ,l,γ,c∈ℂ such that ℜμ,ℜσ,ℜl>0,ℜc>ℜγ>0 with p≥0,δ>0 and 0<k<δ+ℜμ, then(8)ddtmtσ−1Eμ,σ,lγ,δ,k,cωtμ;p=tσ−m−1Eμ,σ−m,lγ,δ,k,cωtμ;p,ℜσ>m.
Fractional integral operators are very useful in mathematical inequalities. The authors have established fractional integral inequalities due to different fractional integral operators, see [15–30]. Many authors have used Mittag-Leffler function to define fractional integral operators. The fractional integral operators corresponding to Mittag-Leffler function (4) are defined as follows:Definition 6.
(see [9]). Let ω,μ,σ,l,γ,c∈ℂ, ℜμ,ℜσ,ℜl>0, and ℜc>ℜγ>0 with p≥0, δ>0, and 0<k≤δ+ℜμ. Let f∈L1a,b and x∈a,b. Then, the generalized fractional integral operators containing Mittag-Leffler function are defined by(9)∈μ,σ,l,ω,a+γ,δ,k,cfx;p=∫axx−tσ−1Eμ,σ,lγ,δ,k,cωx−tμ;pftdt,and(10)∈μ,σ,l,ω,b−γ,δ,k,cfx;p=∫xbt−xσ−1Eμ,σ,lγ,δ,k,cωt−xμ;pftdt,where Eμ,σ,lγ,δ,k,c⋅ is the Mittag-Leffler function defined in (6).
From (9) and (10), one can obtain the fractional integral operators defined in [10–14], see Remark 1.4 of [32]. In [20], Farid et al. proved that(11)∈μ,σ,l,ω,a+γ,δ,k,c1x;p=x−aσEμ,σ+1,lγ,δ,k,cwx−aμ;p,and(12)∈μ,τ,l,ω,b−γ,δ,k,c1x;p=b−xτEμ,τ+1,lγ,δ,k,cwb−xμ;p.
We will follow the upcoming notations in the main results:(13)Dσ,ω,a+x;p=ϵμ,σ,l,ω,aγ,δ,k,c1x;p,(14)Dτ,ω,b−x;p=∈μ,τ,l,ω,b−γ,δ,k,c1x;p.
The upcoming section contains the main results which address the bounds of fractional integral operators containing Mittag-Leffler functions for exponentiallys-convex functions. The continuity of such fractional integrals is proved. A Hadamard inequality is established that leads to several Hadamard inequalities for convex, exponentially convex, and s-convex functions. Moreover, the results of papers [31, 32] can be obtained in particular.


## 2. Main Results

Theorem 1.
Letf:a,b⟶ℝ be a real-valued function. If f is positive and exponentially s-convex, then for σ,τ≥1, the following upper bound for generalized integral operators holds:(15)∈μ,σ,l,ω,a+γ,δ,k,cfx;p+∈μ,τ,l,ω,b−γ,δ,k,cfx;p≤faeαa+fxeαxx−aDσ−1,a+x;ps+1+fbeβb+fxeβxb−xDτ−1,b−x;ps+1,x∈a,b,α,β∈ℝ.Proof.
Letx∈a,b. Then, for t∈a,x and σ≥1, we have the following inequality:(16)x−tσ−1Eμ,σ,lγ,δ,k,cωx−tμ;p≤x−aσ−1Eμ,σ,lγ,δ,k,cωx−aμ;p.
Asf is exponentially s-convex, therefore, one can obtain(17)ft≤x−tx−asfaeαa+t−ax−asfxeαx.
By multiplying (13) and (14) and then integrating over a,x, we get(18)∫axx−tσ−1Eμ,α,lγ,δ,k,cωx−tμ;pftdt≤x−aα−1Eμ,σ,lγ,δ,k,cωx−aμ;px−asfaeαa∫axx−tsdt+fxeαx∫axt−asdt.
The left integral operator follows the upcoming inequality:(19)∈μ,σ,l,ω,a+γ,δ,k,cfx;p≤x−aDσ−1,a+x;ps+1faeαa+fxeαx.
Now, on the contrary, fort∈x,b and τ≥1, we have the following inequality:(20)t−xτ−1Eμ,τ,lγ,δ,k,cωt−xμ;p≤b−xτ−1Eμ,τ,lγ,δ,k,cωb−xμ;p.
Again from exponentiallys-convexity of f, we have(21)ft≤t−xb−xsfbeβb+b−tb−xsfxeβx.
By multiplying (20) and (21) and then integrating over x,b, we get(22)∫xbt−xτ−1Eμ,τ,lγ,δ,k,cωt−xμ;pftdt≤b−xτ−1Eμ,τ,lγ,δ,k,cωb−xμ;pb−xsfbeβb∫xbt−xsdt+fxeβx∫xbb−tsdt.
The right integral operator satisfies the following inequality:(23)∈μ,τ,l,ω,b−γ,δ,k,cfx;p≤b−xDτ−1,b−x;ps+1fbeβb+fxeβx.
By adding (19) and (23), required inequality (15) can be obtained.Corollary 1.
If we setσ=τ in (15), then the following inequality is obtained:(24)∈μ,σ,l,ω,a+γ,δ,k,cfx;p+∈μ,σ,l,ω,b−γ,δ,k,cfx;p≤faeαa+fxeαxx−aDσ−1,a+x;ps+1+fbeβb+fxeβxb−xDσ−1,b−x;ps+1,x∈a,b.Corollary 2.
Along with assumption of Theorem1, if f∈L∞a,b, then the following inequality is obtained:(25)∈μ,σ,l,ω,a+γ,δ,k,cfx;p+∈μ,τ,l,ω,b−γ,δ,k,cfx;p≤f∞s+11eαa+1eαxx−aDσ−1,a+x;p+1eβb+1eβxb−xDτ−1,b−x;p.Corollary 3.
Forσ=τ in (25), we get the following result:(26)∈μ,σ,l,ω,a+γ,δ,k,cfx;p+∈μ,σ,l,ω,b−γ,δ,k,cfx;p≤f∞s+11eαa+1eαxx−aDσ−1,a+x;p+1eβb+1eβxb−xDσ−1,b−x;p.Corollary 4.
Fors=1 in (25), we get the following result for exponentially convex functions:(27)∈μ,α,l,ω,a+γ,δ,k,cfx;p+∈μ,β,l,ω,b−γ,δ,k,cfx;p≤f∞21eαa+1eαxx−aDα−1,a+x;p+1eβb+1eβxb−xDβ−1,b−x;p.Theorem 2.
With the assumptions of Theorem1, if f∈L∞a,b, then operators defined in (9) and (10) are bounded and continuous.Proof.
Iff∈L∞a,b, then from (19), we have(28)∈μ,σ,l,ω,a+γ,δ,k,cfx;p≤f∞x−aDσ−1,a+x;ps+11eαa+1eαx≤2f∞b−aDσ−1,a+b;ps+1eαa,that is,(29)∈μ,σ,l,ω,a+γ,δ,k,cfx;p≤Mf∞,where M=2b−aDσ−1,a+b;p/s+1eαa. Therefore, ∈μ,σ,l,ω,a+γ,δ,k,cfx;p is bounded, and also, it is easy to see that it is linear; hence, this is a continuous operator. Also, on the contrary, from (23), one can obtain(30)∈μ,τ,l,ω,b−γ,δ,k,cfx;p≤Kf∞,where K=2b−aDτ−1,b−a;p/s+1eβa. Therefore, ∈μ,τ,l,ω,b−γ,δ,k,cfx;p is bounded and also it is linear, hence continuous.
The next result provides boundedness of the left and the right fractional integrals at an arbitrary point for functions whose derivatives in absolute values are exponentiallys-convex.Theorem 3.
Letf:a,b⟶ℝ be a real-valued function. If f is differentiable and f′ is exponentially s-convex, then for σ,τ≥1, the following fractional integral inequality for generalized integral operators (9) and (10) holds:(31)∈μ,σ+1,l,ω,a+γ,δ,k,cfx;p+∈μ,τ+1,l,ω,b−γ,δ,k,cfx;p−Dσ−1,a+x;pfa+Dτ−1,b−x;pfb≤f′aeαa+f′xeαxx−aDσ−1,a+x;ps+1+f′beβb+f′xeβxb−xDτ−1,b−x;ps+1,x∈a,b.Proof.
Letx∈a,b and t∈a,x; by using exponentially s-convexity of f′, we have(32)f′t≤x−tx−asf′aeαa+t−ax−asf′xeαx.
From (32), one can have(33)f′t≤x−tx−asf′aeαa+t−ax−asf′xeαx.
The product of (16) and (33) gives the following inequality:(34)x−tσ−1Eμ,σ,lγ,δ,k,cωx−tμ;pf′tdt≤x−aσ−1−sEμ,σ,lγ,δ,k,cωx−aμ;pf′aeαax−ts+f′xeαxt−as.
After integrating the above inequality overa,x, we get(35)∫axx−tσ−1Eμ,σ,lγ,δ,k,cωx−tμ;pf′tdt≤x−aσ−1−sEμ,σ,lγ,δ,k,cωx−aμ;pf′aeαa∫axx−tsdt+f′xeαx∫axt−asdt=x−aσEμ,σ,lγ,δ,k,cωx−tμ;ps+1f′aeαa+f′xeαx.
The left-hand side of (35) is calculated as follows:(36)∫axx−tσ−1Eμ,σ,lγ,δ,k,cωx−tμ;pf′tdt.
Puttingx−t=z, that is, t=x−z and also utilizing (8), we have(37)∫0x−azσ−1Eμ,σ,lγ,δ,k,cωzμ;pf′x−zdz=x−aσ−1Eμ,σ,lγ,δ,k,cωx−aμ;pfa−∫0x−azσ−2Eμ,σ,lγ,δ,k,cωzμ;pfx−zdz.
Now, puttingx−z=t in the second term of the right-hand side of the above equation and then using (9), we get(38)∫0x−azσ−1Eμ,σ,lγ,δ,k,cωzμ;pf′x−zdz=x−aσ−1Eμ,σ,lγ,δ,k,cωx−aμ;pfa−∈μ,σ+1,l,ω,a+γ,δ,k,cfx;p.
Therefore, (35) takes the following form:(39)Dσ−1,a+x;pfa−∈μ,σ+1,l,ω,a+γ,δ,k,cfx;p≤x−aDσ−1,a+x;ps+1f′aeαa+f′xeαx.
Also, from (32), one can have(40)f′t≥−x−tx−asf′aeαa+t−ax−asf′xeαx.
Following the same procedure as we did for (33), one can obtain(41)∈μ,σ+1,l,ω,a+γ,δ,k,cfx;p−Dσ−1,a+x;pfa≤x−aDσ−1,a+x;ps+1f′aeαa+f′xeαx.
From (39) and (41), we get(42)∈μ,σ+1,l,ω,a+γ,δ,k,cfx;p−Dσ−1,a+x;pfa≤x−aDσ−1,a+x;ps+1f′aeαa+f′xeαx.
Now, we letx∈a,b and t∈x,b. Then, by exponentially s-convexity of f′, we have(43)f′t≤t−xb−xsf′beβb+b−tb−xsf′xeβx.
On the same lines as we have done for (16), (33), and (40), one can get from (20) and (43) the following inequality:(44)∈μ,τ+1,l,ω,b−γ,δ,k,cfx;p−Dτ−1,b−x;pfb≤b−xDτ−1,b−x;ps+1f′beβb+f′xeβx.
From inequalities (42) and (44), via triangular inequality, (28) can be obtained.
The following results hold as special cases.Corollary 5.
If we putσ=τ in (28), then the following inequality is obtained:(45)∈μ,σ+1,l,ω,a+γ,δ,k,cfx;p+∈μ,σ+1,l,ω,b−γ,δ,k,cfx;p−Dσ−1,a+x;pfa+Dσ−1,b−x;pfb≤f′aeαa+f′xeαxx−aDσ−1,a+x;ps+1+f′beβb+f′xeβxb−xDσ−1,b−x;ps+1,x∈a,b.Definition 7.
Letf:a,b⟶ℝ be a function; we say f is exponentially symmetric about a+b/2 if(46)fxeαx=fa+b−xeαa+b−x,α∈ℝ.
It is required to give the following lemma which will be helpful to produce Hadamard-type estimations.Lemma 2.
Letf:a,b⟶ℝ be an exponentially s-convex function. If f is exponentially symmetric, then the following inequality holds:(47)fa+b2≤fx2s−1eαx.Proof.
Fora,b⊂ℝ be a closed interval, t∈0,1, and α∈ℝ, we have(48)fa+b2=fat+1−tb2+a1−t+bt2.
Sincef is exponentially s-convex, so(49)fa+b2≤fat+1−tb2seαat+1−tb+fa1−t+bt2seαa1−t+bt.
Letx=at+1−tb, where x∈a,b. Then, we have a+b−x=bt+1−ta, and we get(50)fa+b2≤fx2seαx+fa+b−x2seαa+b−x.
Now, using the fact of exponentially symmetric, we will get (47).Theorem 4.
Letf:a,b⟶ℝ, a>b, be a real-valued function. If f is positive, exponentially s-convex, and symmetric about a+b/2, then for σ,τ>0, the following fractional integral inequality for generalized integral operators (9) and (10) holds:(51)2s−1eαxfa+b2Dτ+1,b−a;p+Dσ+1,a+b;p≤∈μ,τ+1,l,ω,b−γ,δ,k,cfa;p+∈μ,σ+1,l,ω,a+γ,δ,k,cfb;p≤Dτ−1,b−a;p+Dσ−1,a+b;pb−a2s+1faeαa+fbeβb.Proof.
Forx∈a,b, we have(52)x−aτEμ,τ,lγ,δ,k,cωx−aμ;p≤b−aτEμ,τ,lγ,δ,k,cωb−aμ;p,τ>0.
Asf is exponentially s-convex, so for x∈a,b, we have(53)fx≤x−ab−asfbeβb+b−xb−asfaeαa.
By multiplying (52) and (53) and then integrating over a,b, we get(54)∫abx−aτEμ,τ,lγ,δ,k,cωx−aμ;pfxdx≤b−aτ−sEμ,τ,lγ,δ,k,cωb−aμ;pfbeβb∫abx−asdx+faeαa∫abb−xsdx,from which we have(55)∈μ,τ+1,l,ω,b−γ,δ,k,cfa;p≤b−aτ+1Eμ,τ,lγ,δ,k,cωb−aμ;ps+1faeαa+fbeβb,(56)∈μ,τ+1,l,ω,b−γ,δ,k,cfa;p≤b−a2s+1Dτ−1,b−a;pfaeαa+fbeβb.
Now, on the contrary, forx∈a,b, we have(57)b−xσEμ,σ,lγ,δ,k,cωb−xμ;p≤b−aσEμ,σ,lγ,δ,k,cωb−aμ;p,α>0.
By multiplying (53) and (57) and then integrating over a,b, we get(58)∫abb−xσEμ,σ,lγ,δ,k,cωb−xμ;pfxdx≤b−aσ−sEμ,σ,lγ,δ,k,cωb−aμ;pfbeβb∫abx−asdx+faeαa∫abb−xsdx,from which we have(59)∈μ,σ+1,l,ω,a+γ,δ,k,cfb;p≤b−aσ+1Eμ,σ,lγ,δ,k,cωb−aμ;ps+1faeαa+fbeβb,(60)∈μ,σ+1,l,ω,a+γ,δ,k,cfb;p≤b−a2s+1Dσ−1,a+b;pfaeαa+fbeβb.
Adding (56) and (60), we get(61)∈μ,τ+1,l,ω,b−γ,δ,k,cfa;p+∈μ,σ+1,l,ω,a+γ,δ,k,cfb;p≤Dτ−1,b−a;p+Dσ−1,a+b;pb−a2s+1faeαa+fbeβb.
Multiplying (47) with x−aτEμ,τ,lγ,δ,k,cωx−aμ;p and integrating over a,b, we get(62)fa+b2∫abx−aτEμ,τ,lγ,δ,k,cωx−aμ;pdx≤12s−1eαx∫abx−aτEμ,τ,lγ,δ,k,cωx−aμ;pfxdx.
By using (10) and (14), we get(63)fa+b2Dτ+1,b−a;p≤12s−1eαx∈μ,τ+1,l,ω,b−γ,δ,k,cfa;p.
Multiplying (47) with b−xσEμ,σ,lγ,δ,k,cωb−xμ;p, integrating over a,b, and also using (6) and (10), we get(64)fa+b2Dσ+1,a+b;p≤12s−1eαx∈μ,σ+1,l,ω,a+γ,δ,k,cfb;p.
By adding (63) and (64), we get(65)2s−1eαxfa+b2Dτ+1,b−a;p+Dσ+1,a+b;p≤∈μ,τ+1,l,ω,b−γ,δ,k,cfa;p+∈μ,σ+1,l,ω,a+γ,δ,k,cfb;p.
By combining (61) and (65), inequality (51) can be obtained.Corollary 6.
If we putσ=τ in (51), then the following inequality is obtained:(66)2s−1eαxfa+b2Dσ+1,b−a;p+Dσ+1,a+b;p≤∈μ,σ+1,l,ω,b−γ,δ,k,cfa;p+∈μ,σ+1,l,ω,a+γ,δ,k,cfb;p≤Dσ−1,b−a;p+Dσ−1,a+b;pb−a2s+1faeαa+fbeβb.


## 3. Concluding Remarks

We have established the general fractional integral inequalities by using exponentiallys-convex functions. By selecting particular values at the place of parameters, a variety of results can be obtained. For example, the reader can obtain bounds for fractional integral operators defined by Salim and Faraj in [12] by selecting p=0, bounds for fractional integral operators defined by Rahman et al. in [11] by selecting l=δ=1, bounds for fractional integral operators defined by Shukla and Prajapati in [13] by selecting p=0 and l=δ=1, bounds for fractional integral operators defined by Prabhakar in [10] by selecting p=0 and l=δ=k=1, and bounds for Riemann–Liouville fractional integrals by selecting p=ω=0.



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