Atiq Ur Rehman

This paper investigates the class of interval‐valued s ‐convex functions in the second sense using Caputo–Fabrizio integrals. Some generalizations of the Hermite–Hadamard ‐type, Hermite–Hadamard–Fejér ‐type, and Hermite–Hadamard–Mercer ‐type inclusions involving the integrals are developed. While the Jensen’s inclusion and Jensen–Mercer‐type inclusion are also formulated for the same class of functions to help developing ‐type inclusion. Additionally, some inclusions for the product of these functions involving these operators are also presented. Substitutions are applied to the main results, yielding corresponding inequalities for real‐valued s ‐convex functions in the second sense and inequalities and inclusions for real‐valued convex and functions, respectively. These results might open the path for new avenues in modeling, optimization problems, interval differential equations, and fuzzy functions that involve both discrete and continuous variables simultaneously.

Fractional calculus is unique due to the fact it is as old as regular (integer) calculus, but it has also expanded its applications in a variety of fields and on a diversity of topics over the course of the last century. This leads to a continuous increase in the number of researchers and papers, ranging from integral inequality to biological models and then to systems with delay, neutrals, hybrids, etc. In this interplay between experts from various fields and mathematicians themselves, who employ these techniques in their theoretical research, the applications have increased. This research study focuses on the Caputo–Fabrizio operators because of its numerous applications. A set of generalized Hermite–Hadamard‐type inequalities for the Caputo–Fabrizio integrals is constructed via ( α , s , m )‐convex function. For the trapezoidal formula, errors are estimated. Moreover, the disparity between the expected (average) hydrogen ion concentration and pH is estimated in the present investigation. The accuracy of main conclusion is further affirmed by providing examples and graphical representations. In the framework of integral inequalities, this paper shows how the findings strengthen and illuminate a significant amount of earlier work.

The inequality [ ] is well known in literature as Ostrowski inequality. This paper aims to find two fractional versions of this inequality by utilizing definitions of Riemann–Liouville fractional integrals. By applying these inequalities on some fixed points, error bounds of fractional Hermite–Hadamard inequalities are obtained. Further, an Ostrowski–Grüss type inequality is also proved.

In this paper we investigate Chebyshev’s type inequalities for h-convex functions. These inequalities are obtained by imposing some convenient conditions on h-convex functions. Furthermore, the associated Chebyshev’s functional are estimated via mean value theorems.

In this article, we consider the class of modified convex functions and derive the famous Giaccardi and Petrovi type inequalities for this class of functions. The mean value theorems for the functionals due to Giaccardi and Petrovi type inequalities are formulated. Some special cases are discussed by taking different examples of function .

In this paper, a well‐known inequality called Giaccardi inequality is established for isotonic linear functionals by applying s‐convexity in the second sense, which leads to notable Petrović inequality. As a special case, discrete and integral versions of Giaccardi inequality are derived along with the Petrović inequality as a particular case. In application point of view, newly established inequalities are derived for different time scales.

In this article, we prove some fractional versions of Hadamard-type inequalities for strongly exponentially α , h − m -convex functions via generalized Riemann–Liouville fractional integrals. The outcomes of this paper provide inequalities of strongly convex, strongly m -convex, strongly s -convex, strongly α , m -convex, strongly s , m -convex, strongly h − m -convex, strongly α , h − m -convex, strongly exponentially convex, strongly exponentially m -convex, strongly exponentially s -convex, strongly exponentially s , m -convex, strongly exponentially h − m -convex, and exponentially α , h − m -convex functions. The error estimations are also studied by applying two fractional integral identities.

Kragujevac Journal of Mathematics is a periodical for mathematics. Requirements for acceptance of papers include originality, scientific significance and clarity of presentation.

In this paper some new general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function are presented. These results produce inequalities for several kinds of fractional integral operators. Some interesting special cases of our main results are also pointed out.