Abstract
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, firstly we introduce the notion of h -exponential convex functions. This notion can be considered as generalizations of many existing definitions 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.
Highlights
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
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
Summary
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]. 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. Bai et al [10] presented Hermite-Hadamard type inequalities for the m and ðα, mÞ-logarithmically convex functions. Chu et al [12] gave generalizations of Hermite-Hadamard type inequalities for MT-convex functions. Fractional calculus sets new trends in inequalities of convex analysis. We will deal with incomplete gamma functions. We establish some well-known inequalities for the proposed notions via incomplete gamma functions.
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