MONOTONICITY AND CONVEXITY FOR NABLA FRACTIONAL q-DIFFERENCES

In this paper, we examine the relation between monotonicity and convexity for nabla fractional q-differences. In particular we prove that

Theorem A. Assume $f : q^{N_0} \longrightarrow R$, $\nabla^{\nu}_q f(t) ≥ 0$ for each $t \in q^{N_0}$, with $1 < \nu < 2$, then $\nabla_q f(t) \geq 0$ for $t\in q^{N_1}$.

Theorem B. Assume $f : q^{N_0} \longrightarrow R$, $\nabla^{\nu}_q f(t) ≥ 0$ for each $t \in q^{N_1}$, with $2 < \nu < 3$, then $\nabla_q^2 f(t) \geq 0$ for $t\in q^{N_2}$.

This shows that, in some sense, the positivity of the $\mu$-th order $q$-fractional difference has a strong connection to the monotonicity and convexity of $f(t)$.