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Convex Functions on Discrete Time Domains

Published online by Cambridge University Press:  20 November 2018

Ferhan M. Atıcı  and
Hatice Yaldız
Ferhan M. Atıcı
Affiliation:
Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101-3576, USA e-mail: [email protected]
Hatice Yaldız
Affiliation:
Department of Mathematics, Düzce University, Düzce, Turkey e-mail: [email protected]
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Abstract

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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 ${{\Delta }^{2}}f\,\ge \,0$ 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

26B2526A3339A1239A7026E7026D0726D1026D15discrete calculusdiscrete fractional calculusconvex functionsdiscrete Hermite–Hadamard inequality
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 225 - 233
Copyright
Copyright © Canadian Mathematical Society 2016

References

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