Boundedness of Fractional Integral Operators Containing Mittag-Leffler Function via Exponentially s-Convex Functions

East University of Heilongjiang, Department of Basic Course, Harbin, Heilongjiang 150066, China Department of Mathematics, COMSATS University Islamabad, Attock Campus, Islamabad, Pakistan Department of Mathematics, Government College University, Lahore, Pakistan Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Islamabad, Pakistan RUDN University, Moscow, Russia School of Economics and Management, Harbin Engineering University, Harbin, Heilongjiang 150001, China College of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China


Introduction
Convex functions play an important role in many areas of mathematics. ey are important especially in the study of optimization problems, theory of inequalities, mathematical analysis, statistical analysis, operation research, and so on. e 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 function f: K ⊆ R ⟶ R, where K is an interval in R, is said to be convex function if the following inequality holds: for all a, b ∈ K and t ∈ [0, 1].
Definition 2 (see [2]). A function f: K ⊆ R ⟶ R, where K is an interval in R, is said to be exponentially convex function if holds 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 the s-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 holds 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 holds 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][5][6][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. e Mittag-Leffler function denoted by E σ (t) was introduced by Mittag-Leffler [8] in 1903: where σ, t ∈ C, Γ(·) is the gamma function, and R(σ) ≥ 0. e Mittag-Leffler function is a direct generalization of the exponential function to which it reduces for σ � 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. Let μ, σ, l, c, c ∈ C, R(μ), R(σ), R(l) > 0, and R(c) > R(c) > 0 with p ≥ 0, δ > 0, and 0 < k ≤ δ + R(μ).
e 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.

Main Results
If f is positive and exponentially s-convex, then for σ, τ ≥ 1, the following upper bound for generalized integral operators holds:

Journal of Mathematics
Proof. Let x ∈ [a, b]. en, for t ∈ [a, x) and σ ≥ 1, we have the following inequality: As f is exponentially s-convex, therefore, one can obtain By multiplying (13) and (14) and then integrating over e left integral operator follows the upcoming inequality: Now, on the contrary, for t ∈ (x, b] and τ ≥ 1, we have the following inequality: Again from exponentially s-convexity of f, we have By multiplying (20) and (21) and then integrating over e right integral operator satisfies the following inequality: By adding (19) and (23), required inequality (15) can be obtained. □ Corollary 1. If we set σ � τ in (15), then the following inequality is obtained: Corollary 2. Along with assumption of eorem 1, if f ∈ L ∞ [a, b], then the following inequality is obtained: Corollary 3. For σ � τ in (25), we get the following result: Corollary 4. For s � 1 in (25), we get the following result for exponentially convex functions: Theorem 2. With the assumptions of eorem 1, if f ∈ L ∞ [a, b], then operators defined in (9) and (10) are bounded and continuous. Proof that is, where M � (2(b − a)D σ− 1,a + (b; p)/(s + 1)e αa ). erefore, (∈ c,δ,k,c μ,σ,l,ω,a + f)(x; 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 where is bounded and also it is linear, hence continuous. e next result provides boundedness of the left and the right fractional integrals at an arbitrary point for functions whose derivatives in absolute values are exponentially s-convex. □ Theorem 3. Let f: [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: Proof. Let x ∈ [a, b] and t ∈ [a, x]; by using exponentially s-convexity of |f ′ |, we have From (32), one can have e product of (16) and (33) gives the following inequality: (34) After integrating the above inequality over [a, x], we get e left-hand side of (35) is calculated as follows:

(37)
Now, putting x − z � t in the second term of the righthand side of the above equation and then using (9), we get (38) erefore, (35) takes the following form: Also, from (32), one can have Following the same procedure as we did for (33), one can obtain 4 Journal of Mathematics From (39) and (41), we get en, by exponentially s-convexity of |f ′ |, we have On the same lines as we have done for (16), (33), and (40), one can get from (20) and (43) the following inequality: From inequalities (42) and (44), via triangular inequality, (28) can be obtained. e following results hold as special cases.
□ Corollary 5. If we put σ � τ in (28), then the following inequality is obtained: It is required to give the following lemma which will be helpful to produce Hadamard-type estimations.

Lemma 2. Let f: [a, b] ⟶ R be an exponentially s-convex
function. If f is exponentially symmetric, then the following inequality holds: Proof. For [a, b] ⊂ R be a closed interval, t ∈ [0, 1], and α ∈ R, we have Since f is exponentially s-convex, so Now, using the fact of exponentially symmetric, we will get (47).
As f is exponentially s-convex, so for x ∈ [a, b], we have By multiplying (52) and (53) and then integrating over Journal of Mathematics 5 from which we have Now, on the contrary, for x ∈ [a, b], we have By multiplying (53) and (57) and then integrating over from which we have Adding (56) and (60), we get Multiplying (47) with (x − a) τ E c,δ,k,c μ,τ,l (ω(x − a) μ ; p) and integrating over [a, b], we get By using (10) and (14), we get c,δ,k,c μ,τ+1,l,ω,b − f (a; p).

Concluding Remarks
We have established the general fractional integral inequalities by using exponentially s-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.