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 $f(t)$.

}, keywords = {26A33, 26A48, 39A70, 39A99.}, issn = {1056-2176}, url = {https://www.acadsol.eu/dsa/articles/25/4.pdf}, author = {JIA BAOGUO and LYNN ERBE and ALLAN PETERSON} }