Hermite-Hadamard’s Inequality on Time Scales

1. Introduction

Hermite-Hadamard's inequality is of great interest in differential and difference equations, and other areas of mathematics. In 1983, Hermite published the renowned inequality as follows:if is convex. Hermite's result was mentioned in the form of Hartman (1992) and Mitrinović (1970), but it was not mentioned that this inequality is due to Hermite. Since Mitrinović (1965, 1970, 1972) mentioned this inequality and called it Hadamard's inequality, many mathematicians have discussed inequality , (Alzer, 1989; Agarwal, Bohner, & Peterson, 2001; Bohner & Peterson, 2001; Brenner & Alzer, 1991; Dragomir, 1992, 2000; Yang & Hong, 1997; Yang & Tseng, 1990). For further details on his history, see Mitrinović and Lacković (1985). The purpose of the paper is to establish Hermite-Hadamard's inequality in time scale version.

Now, we briefly introduce the time scales calculus as follows.

A time scale is a closed subset of the set . We assume that any time scale has the topology that it inherits from the standard topology on . Since a time scale may or may not be connected, we need the concept of jump operators.

• Definition 1.A. Let , where is a time scale, we define the forward and backward jump operators by

  

A point , , is said to be right-scattered if , left-scattered if , right-dense if , left-dense if .