On Convexity, Monotonicity and Positivity Analysis for Discrete Fractional Operators Defined Using Exponential Kernels

Abstract

This article deals with analysing the positivity, monotonicity and convexity of the discrete nabla fractional operators with exponential kernels from the sense of Riemann and Caputo operators. These operators are called discrete nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional operators. Further, some of our results concern sequential nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional differences, such as ∇aCFRμ∇bCFCυh(x), for various values of start points a and b, and for orders υ and μ in different ranges. Three illustrative examples of the main lemmas in the case of Riemann–Liouville are given at the end of the article.