Novel Analysis of Hermite–Hadamard Type Integral Inequalities via Generalized Exponential Type m-Convex Functions

In this present case, we focus and explore the idea of a new family of convex function namely exponentialtype m–convex functions. To support this newly introduced idea, we elaborate some of its nice algebraicproperties. Employing this, we investigate the novel version of Hermite–Hadamard type integral inequality.Furthermore, to enhance the paper, we present several new refinements of Hermite–Hadamard (H−H) inequality.Further, in the manner of this newly introduced idea, we investigate some applications of specialmeans. These new results yield us some generalizations of the prior results in the literature. We believe, themethodology investigated in this paper will further inspire intrigued researchers.

In this paper, we propose some generalized integral inequalities of the Raina type depicting the Mittag–Leffler function. We introduce and explore the idea of generalized s-type convex function of Raina type. Based on this, we discuss its algebraic properties and establish the novel version of Hermite–Hadamard inequality. Furthermore, to improve our results, we explore two new equalities, and employing these we present some refinements of the Hermite–Hadamard-type inequality. A few remarkable cases are discussed, which can be seen as valuable applications. Applications of some of our presented results to special means are given as well. An endeavor is made to introduce an almost thorough rundown of references concerning the Mittag–Leffler functions and the Raina functions to make the readers acquainted with the current pattern of emerging research in various fields including Mittag–Leffler and Raina type functions. Results established in this paper can be viewed as a significant improvement of previously known results.

In this work, we introduce the idea of n –polynomial harmonically s –type convex function. We elaborate the new introduced idea by examples and some interesting algebraic properties. As a result, new Hermite–Hadamard, some refinements of Hermite–Hadamard and Ostrowski type integral inequalities are established, which are the generalized variants of the previously known results for harmonically convex functions. Finally, we illustrate the applicability of this new investigation in special functions (hypergeometric function and special mean of real numbers).

In this paper, we define and investigate generalized exponential type convex functions namely exponentially $s$--convex function. In the support of this newly introduced idea, we attain the algebraic properties of this function, and furthermore, in the frame of simple calculus, we explore and attain the novel kind of Ostrowski type inequalities.

In the paper, with the help of two known integral identities and by virtue of the classical Hölder integral inequality, the authors establish several new integral inequalities of the Hermite–Hadamard type for convex functions. These newly established inequalities generalize some known results.

The main objective of this paper is basically to acquire some new extensions of Ostrowski type inequalities for the function whose first derivatives' absolute value are $s$--type $p$--convex. We initially presented a new auxiliary definition namely $s$--type $p$--convex function. Some beautiful algebraic properties and examples related to the newly introduced definition are discussed. We additionally investigated some beautiful cases that can be derived from the novel refinements of the paper. These new results yield us some generalizations of the prior results. We trust that the techniques introduced in this paper will further motivate intrigued researchers.

In this work, we introduce the idea and concept of m –polynomial p –harmonic exponential type convex functions. In addition, we elaborate the newly introduced idea by examples and some interesting algebraic properties. As a result, several new integral inequalities are established. Finally, we investigate some applications for means. The amazing techniques and wonderful ideas of this work may excite and motivate for further activities and research in the different areas of science.

AbstractIn this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space {\mathbb{R}^{n}} for any {n\geq 2}.

AbstractA specific type of convex functions is discussed. By examining this, we investigate new Hermite–Hadamard type integral inequalities for the Riemann–Liouville fractional operators involving the generalized incomplete gamma functions. Finally, we expose some examples of special functions to support the usefulness and effectiveness of our results.