On Extended Convex Functions via Incomplete Gamma Functions

Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. In this paper, ﬁ rstly we introduce the notion of h -exponential convex functions. This notion can be considered as generalizations of many existing de ﬁ nitions of convex functions. Then, we establish some well-known inequalities for the proposed notion via incomplete gamma functions. Precisely speaking, we established trapezoidal, midpoint, and He ’ s inequalities for h -exponential and harmonically exponential convex functions via incomplete gamma functions. Moreover, we gave several remarks to prove that our results are more generalized than the existing results in the literature.


Introduction
Convex optimization contributed largely in many areas of pure and applied mathematics during recent years, and convex analysis provides main foundation for convex optimization [1,2]. Due to huge applications of convex analysis, the researchers always show interest to generalization the notion of convexity. In literature, there exist many versions of convex functions, for example, h-convex function, see [3], r-convex functions, see [4], harmonic convex function, see [5], exponentially convex functions, see [6], etc. [7,8].
Since convex function is a class of very important functions which is widely used in pure mathematics, functional analysis, optimization theory, and mathematical economics, so to study properties of certain classes of convex functions and establish different inequalities like trapezoidal, midpoint, He's Hermite-Hadamard, Fejér, etc., type inequality is an important area of research. A lot of work is devoted to establish different kinds of inequalities for different classes of convex functions, for example, Iscan [9] established Hermite-Hadamard type inequalities for harmonically convex functions. Bai et al. [10] presented Hermite-Hadamard type inequalities for the m and ðα, mÞ-logarithmically convex functions. Özdemir et al. [11] developed Hermite-Hadamard-type inequalities via ðα, mÞ-convex functions. Chu et al. [12] gave generalizations of Hermite-Hadamard type inequalities for MT-convex functions.
It is always appreciable to derive more version of inequalities for generalized convexities. For some important generalization, we refer [13,14]. Fractional calculus also provides some broader variety to deal real-world problems. Just like other fields, fractional calculus also sets new trends in inequalities of convex analysis. For more details on fractional integral inequalities, we refer to the readers [15][16][17][18]. Many interesting controversies are also part of history of fractional calculus. Some famous definitions of fractional derivative are Riemann-Liouville [19], Caputo-Fabrizio [20], etc. [21][22][23][24]. In the present paper, we will deal with incomplete gamma functions. Firstly, we introduce the notions of h-exponential convex functions and harmonically exponential convex functions. Then, we establish some well-known inequalities for the proposed notions via incomplete gamma functions. Precisely speaking, we established trapezoidal, midpoint, and He's inequalities for h-exponential and harmonically exponential convex functions via incomplete gamma functions. Moreover, we gave several remarks to prove that our results are more generalized than the existing results in the literature.
The breakup of this paper is as follows: In Section 2, we present basic definitions and known results. Section 3 contains trapezoidal type inequalities via incomplete gamma function. Midpoint inequalities via incomplete gamma function are presented in Section 4, and He's inequality via the incomplete gamma functions is presented in Section 5. Last section contains concluding remarks and some future directions.

Preliminaries
Before starting the main findings, we review some definitions, notations, and theorems which are necessary to proceed. Throughout this paper, L 1 denotes space of all locally integrable functions.
Definition 1 [19]. For any L 1 function zðuÞ on an interval ½x, y with u ∈ ½x, yk-th left-RL fractional integral of zðuÞ is given by for Re ðkÞ > 0. Also, the k-th right-RL fractional integral of zðuÞ is given by Definition 2 [6]. We say that the function z : holds for every x, y ∈ M and t ∈ ½0, 1: . We say that the function z: where x, y ∈ M and t ∈ ½0, 1: We are now ready to define some new convexity, called as h-exponential convex function.
Definition 4. We say that the function z: where x, y ∈ M and t ∈ ½0, 1: (1) By substituting hðe t − 1Þ = 1/ðe t − 1Þ, hðe 1−t − 1Þ = 1/ðe 1−t − 1 in Definition 3, we get harmonically exponential convex function in Definition 3, we get Definition 2 of exponential convex function Now, the integral inequality of Hermite-Hadamard (HH) type for a convex function is give by Sarikaya et al. [25] generalized the HH-inequality (6) to fractional integrals of RL type which is given by where k > 0 and z½x, y ⟶ R is let to be an L 1 convex function. After that, Sarikaya and Yildirim [26] found a new inequality of the above The following facts will be needed in establishing our main results: Remark 6 (21). For Re > 0, the following identities hold: Remark 7 (21). For Re > 0, the following identities hold: 2 Journal of Function Spaces Lemma 8 [25]. If z : ½x, y ⟶ R is L 1 ½x, y with 0 < x < y and k > 0, then we have Lemma 9 [26]. If z : ½x, y ⟶ R is L 1 ½x, y with 0 < x < y and k > 0, then we have

Trapezoidal Type Inequalities via Incomplete Gamma Function
In this section, we present trapezoidal type inequalities via incomplete gamma function.
Proof. Let z : I ⟶ R is h-exp convex function and k > 0 then by definition Multiplying t k−1 on both sides and then integrating on ½0, 1, we get Again by small substitution, we have For other inequalities, take Adding both inequalities, we get Multiplying both sides by t k−1 and integrating on [0,1], we have By making the change of variables

Journal of Function Spaces
Multiplying by k > 0 and hðe 1/2 − 1Þ on both sides, we get.
Theorem 13. Let z : ½x, y ⟶ R be L 1 ½x, y with 0 < x < y and k > 0. If |z | is an h-exp convex function, then we where Proof. From Lemma 8, we have By using the h-exp convexity of jzj By using identities (25), we get required result. ☐

Midpoint Inequalities via Incomplete Gamma Function
This section contains midpoint inequalities via incomplete gamma function.

Theorem 14.
Let z : ½x, y ⟶ R be L 1 ½x, y with 0 < x < y and k > 0. If jzj is an h -exp convex function, then we Journal of Function Spaces Multiplying by t k−1 on both sides and integrating w.r.t "t" from [0,1], we get Again by small substitution, we have For other inequalities, take