DISCRETE HERMITE–HADAMARD-TYPE INEQUALITIES FOR (s,m)-CONVEX FUNCTION

Abstract

The Hermite–Hadamard (HH)-type inequality plays a very important role in the fields of basic mathematics and applied mathematics. In recent years, many scholars have expanded and improved it. Although we have achieved some research results about HH-type inequality, the research on discrete HH-type inequalities has just begun, and a lot of work needs to be improved. In this paper, we introduce [Formula: see text]-convex function and present discrete HH-type inequalities on time scale with discrete substitution method. In addition, the Hermite–Hadamard–Fejér(HHF)-type inequalities on time scale will be obtained, where the integrand is [Formula: see text], [Formula: see text] is [Formula: see text]-convex function on [Formula: see text] and [Formula: see text] is symmetric with respect to [Formula: see text], our results in some special cases yield the well-known classic HHF-type inequalities. Finally, through the discrete substitution method, we get discrete fractional HH-type inequality and discrete fractional HHF-type inequality for [Formula: see text]-convex function.