Abstract view
Convex Functions on Discrete Time Domains


Published:20160203
Printed: Jun 2016
Ferhan M. Atıcı,
Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 421013576, USA
Hatice Yaldız,
Department of Mathematics, Düzce University, Düzce, Turkey
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 HermiteHadamard inequality using
the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$).
Second, we state and prove the discrete fractional HermiteHadamard
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.
MSC Classifications: 
26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15 show english descriptions
Convexity, generalizations Fractional derivatives and integrals Discrete version of topics in analysis Difference operators [See also 47B39] Real analysis on time scales or measure chains {For dynamic equations on time scales or measure chains see 34N05} Inequalities involving other types of functions Inequalities involving derivatives and differential and integral operators Inequalities for sums, series and integrals
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
