Hermite-Hadamard’s Inequality on Time Scales

Hermite-Hadamard’s Inequality on Time Scales

Fu-Hsiang Wong (National Taipei University of Education, Taiwan), Wei-Cheng Lian (National Kaohsiung Marine University, Taiwan), Cheh-Chih Yeh (Lunghwa University of Science and Technology, Taiwan) and Ruo-Lan Liang (National Taipei University of Education, Taiwan)
Copyright: © 2011 |Pages: 8
DOI: 10.4018/ijalr.2011070106
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We establish several Hermite-Hadamard’s inequalities on time scales.
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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:jalr.2011070106.m08jalr.2011070106.m09if jalr.2011070106.m10 is convex. Hermite's result was mentioned in the form jalr.2011070106.m11 of Hartman (1992) and Mitrinović (1970), but it was not mentioned that this inequality jalr.2011070106.m12 is due to Hermite. Since Mitrinović (1965, 1970, 1972) mentioned this inequality jalr.2011070106.m13 and called it Hadamard's inequality, many mathematicians have discussed inequality jalr.2011070106.m14, (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 jalr.2011070106.m15 in time scale version.

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

A time scale jalr.2011070106.m16 is a closed subset of the set jalr.2011070106.m17. We assume that any time scale has the topology that it inherits from the standard topology on jalr.2011070106.m18. Since a time scale may or may not be connected, we need the concept of jump operators.

  • Definition 1.A. Let jalr.2011070106.m19, where jalr.2011070106.m20 is a time scale, we define the forward and backward jump operators by


where jalr.2011070106.m22 and jalr.2011070106.m23.

A point jalr.2011070106.m24, jalr.2011070106.m25, is said to be right-scattered if jalr.2011070106.m26, left-scattered if jalr.2011070106.m27, right-dense if jalr.2011070106.m28, left-dense if jalr.2011070106.m29.

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