Convex Functions on Discrete Time Domains

Canadian mathematical bulletin

Ferhan M. Atıcı¹  Hatice Yaldız² 

[1] Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101-3576, USA;Department of Mathematics, Düzce University, Düzce, Turkey

关键词: discrete calculus;    discrete fractional calculus;    convex functions;    discrete Hermite-Hadamard inequality;   

DOI  :  10.4153/CMB-2015-065-6

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

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 valuedfunction on any time scale.

【 授权许可】

Unknown