Search

The aim of this article is to present fractional versions of the Hadamard type inequalities for exponentially $(s, m)$-convex functions via $k$-analogue of Riemann-Liouville fractional integrals. The results provide generalizations of various known fractional integral inequalities. Some special cases are analyzed in the form of corollaries and remarks.

The stochastic processes is a significant branch of probability theory, treating probabilistic models that develop in time. It is a part of mathematics, beginning with the axioms of probability and containing a rich and captivating arrangement of results following from those axioms. In probability, a convex function applied to the expected value of an random variable is always bounded above by the expected value of the convex function of the random variable. The definition of η-convex stochastic process is introduced in this paper. Moreover some basic properties of η-convex stochastic process are derived. We also derived Jensen, Hermite–Hadamard and Ostrowski type inequalities for η-convex stochastic process.

Abstract This article investigates new inequalities for generalized Riemann–Liouville fractional integrals via the refined ( α , h − m ) $(\alpha ,h-m)$ -convex function. The established results give refinements of fractional integral inequalities for ( h − m ) $(h-m)$ -convex, ( α , m ) $(\alpha ,m)$ -convex, ( s , m ) $(s,m)$ -convex, and related functions. Also, the k-fractional versions of given inequalities by using a parameter substitution are provided.

In this paper Hadamard type inequalities for strongly (α,m)-convex functions via generalized Riemann-Liouville fractional integrals are studied. These inequalities provide generalizations as well as refinements of several well known inequalities. The established results are further connected with fractional integral inequalities for Riemann-Liouville fractional integrals of convex, strongly convex and strongly m-convex functions. By using two fractional integral identities some more Hadamard type inequalities are proved.

In this paper we give Hadamard inequalities for exponentially (α,h−m)-convex functions using Riemann-Liouville fractional integrals for strictly increasing function. Results for Riemann-Liouville fractional integrals of convex, m-convex, s-convex, (α,m)-convex, (s,m)-convex, (h−m)-convex, (α,h−m)-convex, exponentially convex, exponentially m-convex, exponentially s-convex, exponentially (s,m)-convex, exponentially (h−m)-convex, exponentially (α,h−m)-convex functions are particular cases of the results of this paper. The error estimations of these inequalities by using two fractional integral identities are also given.

Recently, a generalization of convex function called exponentially (α,h−m)-convex function has been introduced. This generalization of convexity is used to obtain upper bounds of fractional integral operators involving Mittag-Leffler (ML) functions. Moreover, the upper bounds of left and right integrals lead to their boundedness and continuity. A modulus inequality is established for differentiable functions. The Hadamard type inequality is proved which shows upper and lower bounds of sum of left and right sided fractional integral operators.

In this paper we study the Fejér-Hadamard inequalities for convex function with respect to a strictly monotone function. We establish two inequalities for convex function with respect to a strictly monotone function via Riemann-Liouville fractional integrals. From inequalities found here many new results can be derived by selecting specific strictly monotone and weight functions. Also a variety of existing Fejér-Hadamard and Hadamard inequalities can be reproduced.

Abstract In this paper we are presenting the refinements of integral inequalities established for convex functions. Consequently, we get refinements of several fractional integral inequalities for different kinds of fractional integral operators.

The main objective of this paper is to compute refinements of bounds of the generalized fractional integral operators containing an extended generalized Mittag-Leffler function in their kernels. The presented results also provide refinements of already known bounds of different fractional integral operators for convex, m-convex, s-convex and (s, m)-convex functions. Moreover, the refinements of some known fractional versions of the Hadamard inequality are given.

In this article, generalized versions of the k-fractional Hadamard and Fejér-Hadamard inequalities are constructed. To obtain the generalized versions of these inequalities, k-fractional integral operators including the well-known Mittag-Leffler function are utilized. The class of (p,h)-convex functions for Hadamard-type inequalities give the generalizations of results which have been proved in literature for p-convex, h-convex, and several functions deducible from these two classes.

In this paper, we study integral inequalities which will provide refinements of bounds of unified integral operators established for convex and α,m-convex functions. A new definition of function, namely, strongly α,m-convex function is applied in different forms and an extended Mittag-Leffler function is utilized to get the required results. Moreover, the obtained results in special cases give refinements of fractional integral inequalities published in this decade.

In this paper, generalized versions of Hadamard and Fejér–Hadamard type fractional integral inequalities are obtained. By using generalized fractional integrals containing Mittag-Leffler functions, some well-known results for convex and harmonically convex functions are generalized. The results of this paper are connected with various published fractional integral inequalities.

This paper aims to obtain the bounds of a class of integral operators containing Mittag–Leffler functions in their kernels. A recently defined unified Mittag–Leffler function plays a vital role in connecting the results of this paper with the well-known bounds of fractional integral operators published in the recent past. The symmetry of a function about a line is a fascinating property that plays an important role in mathematical inequalities. A variant of the Hermite–Hadamard inequality is established using the closely symmetric property for (α,m)-convex functions.

Fractional integral operators are useful tools for generalizing classical integral inequalities. Convex functions play very important role in the theory of mathematical inequalities. This paper aims to investigate the Hadamard type inequalities for a generalized class of functions namely strongly (α,h−m)-p-convex functions by using Riemann–Liouville fractional integrals. The results established in this paper give refinements of various well-known inequalities which have been published in the recent past.

In the present note, we will introduce the definition of generalized p convex function. We will investigate some properties of generalized p convex function. Moreover, we will develop Jensen’s type, Schur type, and Hermite Hadamard type inequalities for generalized p convex function.

The aim of this paper is to introduce the definition of a generalized strongly p-convex function for higher order. We will develop some basic results related to generalized strongly p-convex function of higher order. Moreover, we will develop Hermite–Hadamard-, Fejér-, and Schur-type inequalities for this generalization.

In this paper, we define a new function, namely, harmonically α,h−m-convex function, which unifies various kinds of harmonically convex functions. Generalized versions of the Hadamard and the Fejér–Hadamard fractional integral inequalities for harmonically α,h−m-convex functions via generalized fractional integral operators are proved. From presented results, a series of fractional integral inequalities can be obtained for harmonically convex, harmonically h−m-convex, harmonically α,m-convex, and related functions and for already known fractional integral operators.

The class of exponentially α,h−m-convex functions has been discovered to unify different kinds of convexities. This paper finds new Riemann–Liouville fractional Hadamard-type inequalities for this generalized class of convex functions. These results further have their consequences which are presented in the form of corollaries. Moreover, some known results are identified in the form of remarks.

In the literature of mathematical inequalities, convex functions of different kinds are used for the extension of classical Hadamard inequality. Fractional integral versions of the Hadamard inequality are also studied extensively by applying Riemann–Liouville fractional integrals. In this article, we define (α,h−m)-convex function with respect to a strictly monotone function that unifies several types of convexities defined in recent past. We establish fractional integral inequalities for this generalized convexity via Riemann–Liouville fractional integrals. The outcomes of this work contain compact formulas for fractional integral inequalities which generate results for different kinds of convex functions.

Mathematical inequalities have gained importance and popularity due to the application of integral operators of different types. The present paper aims to give Chebyshev-type inequalities for generalized k-integral operators involving the Mittag-Leffler function in kernels. Several new results can be deduced for different integral operators, along with Riemann–Liouville fractional integrals by substituting convenient parameters. Moreover, the presented results generalize several already published inequalities.