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# Convex Functions on Discrete Time Domains

Published:2016-02-03
Printed: Jun 2016
• Ferhan M. Atıcı,
Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101-3576, USA
• Hatice Yaldız,
Department of Mathematics, Düzce University, Düzce, Turkey
 Format: LaTeX MathJax PDF

## Abstract

In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f \geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept, we state and prove discrete Hermite-Hadamard inequality using the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$). Second, we state and prove the discrete fractional Hermite-Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale.
 Keywords: discrete calculus, discrete fractional calculus, convex functions, discrete Hermite-Hadamard inequality
 MSC Classifications: 26B25 - Convexity, generalizations 26A33 - Fractional derivatives and integrals 39A12 - Discrete version of topics in analysis 39A70 - Difference operators [See also 47B39] 26E70 - Real analysis on time scales or measure chains {For dynamic equations on time scales or measure chains see 34N05} 26D07 - Inequalities involving other types of functions 26D10 - Inequalities involving derivatives and differential and integral operators 26D15 - Inequalities for sums, series and integrals

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