Some Inequalities for a New Class of Convex Functions with Applications via Local Fractional Integral

Abstract

The understanding of inequalities in convexity is crucial for studying local fractional calculus efficiency in many applied sciences. In the present work, we propose a new class of harmonically convex functions, namely, generalized harmonically ψ - s -convex functions based on fractal set technique for establishing inequalities of Hermite-Hadamard type and certain related variants with respect to the Raina’s function. With the aid of an auxiliary identity correlated with Raina’s function, by generalized Hölder inequality and generalized power mean, generalized midpoint type, Ostrowski type, and trapezoid type inequalities via local fractional integral for generalized harmonically ψ - s -convex functions are apprehended. The proposed technique provides the results by giving some special values for the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

Error estimates of Hermite‐Hadamard type inequalities with respect to a monotonically increasing function

“…Various convex function estimations, expansions, and modifications have been proposed by a number of scholars. Hu et al [15] proposed several inequalities for a new category of convex functions, as well as implications involving the local fractional integral. Omrani et al [16] created a Hermite‐Hadamard inequality for $$$ \left(p,h\right) $$$‐convex functions and also introduced this class.…”

Some new generalizations for exponentially (s, m)-preinvex functions considering generalized fractional integral operators

“…In perspective on the broad utilization of such frameworks, numerous scientists went to the examination of the hypothetical parts of fractional differential conditions. Specifically, there was unique consideration regarding demonstrating the integral inequalities for fractional systems enhanced with an assortment of classical and non-classical (nonlocal) operators with the guide of advance strategies for functional investigation, see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,59,60,61,62,63,64,65]. Fractional integral inequalities involving convex functions played a significant role in the mathematical analysis due to their prominent features and convenient characterizations.…”