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Search: MSC category 26A33 ( Fractional derivatives and integrals )

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Atıcı, Ferhan M.; Yaldız, Hatice
Convex Functions on Discrete Time Domains
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
Categories:26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15

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