Bounds of a Unified Integral Operator via Exponentially ðs,mÞ-Convexity and Their Consequences

Various known fractional and conformable integral operators can be obtained from a unified integral operator. The aim of this paper is to find bounds of this unified integral operator via exponentially ðs,mÞ-convex functions. The resulting bounds provide compact formulas for the bounds of associated fractional and conformable integral operators. Several Hadamard-type inequalities have been produced from a compact version for unified integral operators for exponentially ðs,mÞ-convex functions.

Definition 1 (see [6]). Let f : ½a, b ⟶ ℝ be an integrable function. Also let g be an increasing and positive function on ða, b, having a continuous derivative g ′ on ða, bÞ. The left-sided and right-sided fractional integrals of a function f with respect to another function g on ½a, b of order μ where RðμÞ > 0 are defined by where Γð:Þ is the gamma function.
Recently, a unified integral operator is defined as follows: Definition 4 (see [20]). Let f , g : ½a, b ⟶ ℝ, 0 < a < b, be the functions such that f be positive and f ∈ L 1 ½a, b, and g be differentiable and strictly increasing. Also let ϕ/x be an increasing function on ½a, ∞Þ and α, l, γ, c ∈ ℂ, p, μ, δ ≥0, and 0 < k ≤ δ + μ. Then, for x ∈ ½a, b, the left and right integral operators are defined by For suitable settings of functions ϕ, g, and certain values of parameters included in Mittag-Leffler function (8), some interesting consequences can be obtained which are comprised in the upcoming remarks.
Definition 7. Let s ∈ ½0, 1 and I ⊆ ½0,∞Þ be an interval. A function f : I ⟶ ℝ is said to be exponentially ðs, mÞ-convex in the second sense, if holds for all x, y ∈ I, m ∈ ½0, 1 and α ∈ ℝ: One can note the deducible definitions in the following remark: Remark 8. In the upcoming section, bounds of unified integral operators for exponentially ðs, mÞ-convex functions are given in different forms. Bounds of associated fractional and con-formable integral operators which are known in literature are also deduced. The Hadamard inequality is derived for exponentially ðs, mÞ-convex functions. Its diverse conformable and fractional versions are presented. A modulus inequality is established by using exponentially ðs, mÞ-convexity of |f ′ | . In Section 3, boundedness and continuity of these operators are given.

Main Results
Bounds of unified integral operators (9) and (10), by using exponentially ðs, mÞ-convexity, are established in the following theorem: , be a positive exponentially ðs, mÞ-convex function with m ∈ ð0, 1 and g : ½a, b ⟶ ℝ be a differentiable and strictly increasing function. Also let ϕ/x be an increasing function on ½a, b.

Journal of Function Spaces
Multiplying (15) and (16) and integrating over ½a, x, we can obtain By using Definition 4 and integrating by parts, the following inequality is obtained: Now on the other hand for t ∈ ðx, b and x ∈ ða, bÞ, the following inequality holds true: Using exponentially ðs, mÞ-convexity of f , we have Adopting the same pattern as we did for (15) and (16), we obtained the following inequality from (19) and (20): By adding (18) and (21), (12) can be obtained.
The following result provides the Hadamard inequality for unified integral operators defined in (9)