A monotonicity result for the $q-$fractional operator

Abstract: In this article we prove that if the $q-$fractional operator $(~{q}\nabla{qa}^\alpha y)(t)$ of order $0<\alpha\leq 1$ , $0<q\<1$ and starting at some $qa \in T_q=\{q^k: k \in \mathbb{Z}\}\cup \{0\},~~a\>0$ is positive such that $y(a) \geq 0$, then $y(t)$ is $c_q(\alpha)-$increasing, $c_q(\alpha)=\frac{1-q^\alpha}{1-q}q^{1-\alpha}$. Conversely, if y(t) is increasing and $y(a)\geq 0$, then $(~{q}\nabla{qa}^\alpha y)(t)\geq 0$. As an application, we proved a $q-$fractional version of the Mean-Value Theorem.