Fractional Ostrowski Type Inequalities Via (s, r)−Convex Function

Ali Hassan ⁽¹⁾ and Asif Raza Khan ⁽²⁾

(1) Department of Mathematics, Shah Abdul Latif University Khairpur- 66020, Pakistan

Email address: alihassan.iiui.math@gmail.com

(2) Department of Mathematics, University of Karachi, University Road, Karachi-75270, Pakistan 

Email address: asifrk@uok.edu.pk

Doi : https://doi.org/10.47013/15.4.16

Cited by : Jordan J. Math & Stat., 15 (4B) (2022), 1031 - 1047

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 Abstract. We are introducing very first time a generalized class named it the class of (s, r)−convex functions in mixed kind. This generalized class contains many subclasses including class of s−convex functions in 1st and 2nd kind, P−convex functions, quasi convex functions and the class of ordinary convex functions. Also, we would like to state the generalization of the classical Ostrowski inequality via fractional integrals, which is obtained for functions whose first derivative in absolute values is (s, r)− convex function in mixed kind. Moreover we establish some Ostrowski type inequalities via fractional integrals and their particular cases for the class of functions whose absolute values at certain powers of derivatives are (s, r)−convex functions in mixed kind by using different techniques including Hölder’s inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, some applications in terms of special means would also be given.

Keywords: Ostrowski inequality, convex functions, power mean inequality, Hölder’s inequality.