Monotonicity of sequences involving generalized convexity function and sequences

Abstract

In this paper, by using the theory of generalized convexity functions we introduce and prove monotonicity of sequences of the forms $$ \left\{\left(\prod\limits_{k=1}^nf\left({a_k\over a_n}\right)\right)^{1/n}\right\},\quad \left\{\left(\prod\limits_{k=1}^nf\left({\varphi(k)\over\varphi(n)}\right)\right)^{1/\varphi(n)}\right\}, $$ $$ \left\{{1\over n}\sum_{k=1}^nf\left({a_n\over a_k}\right)\right\}\quad\text{or}\quad \left\{{1\over\varphi(n)}\sum_{k=1}^nf\left({\varphi(n)\over\varphi(k)}\right)\right\}, $$ where $f$ belongs to the classes of $AG$-convex (concave), $HA$-convex (concave), or $HG$-convex (concave) functions defined on suitable intervals, $\{a_n\}$ is a given sequence and $\varphi$ is a given function that satisfy some preset conditions. As a consequence, we obtain some generalizations of Alzer type inequalities.