Two monotonicity results for nabla and delta fractional differences

Abstract

In this paper we study the nabla and delta fractional differences and obtain the following two main results:

Theorem A

If ${f:\mathbb{N}_{a} \to\mathbb{R}}$, ${\nabla^\nu_af(t)\geq 0}$, for each ${t\in\mathbb{N}_{a+1}}$, with ${1 < \nu < 2}$, then ${\nabla f(t)\geq 0}$ for ${t\in\mathbb{N}_{a+2}}$.

Theorem B If ${f:\mathbb{N}_a\to\mathbb{R}}$, ${\Delta^\nu_af(t)\geq 0}$, for each ${t\in\mathbb{N}_{a+2-\nu}}$, with ${1 < \nu < 2}$, and ${f(a+1)\geq f(a)\ge 0}$, then ${\Delta f(t)\geq 0}$ for ${t\in\mathbb{N}_{a}}$.

This demonstrates that there are substantial differences between the nabla fractional difference and the corresponding delta fractional difference. Moreover, we also prove two important inequalities which illustrate that, in some sense, the positivity of the ${\nu}$-th order fractional difference is very closely connected to the monotonicity of f(t).