# MONOTONICITY AND CONVEXITY FOR NABLA FRACTIONAL q-DIFFERENCES

 Title MONOTONICITY AND CONVEXITY FOR NABLA FRACTIONAL q-DIFFERENCES Publication Type Journal Article Year of Publication 2016 Authors BAOGUO JIA, ERBE LYNN, PETERSON ALLAN Journal Dynamic Systems and Applications Volume 25 Start Page 47 Pagination 14 Date Published 2016 ISSN 1056-2176 AMS Subject Classification 26A33, 26A48, 39A70, 39A99. Abstract 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 $f(t)$. PDF https://www.acadsol.eu/dsa/articles/25/4.pdf Refereed Designation Refereed