CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

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 discrete calculus, discrete fractional calculus, convex functions, discrete Hermite-Hadamard inequality
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
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/