Monotonicity of sequences involving generalized convexity function and sequences

Main Article Content

Nguyen Ngoc Hue
Duong Quoc Huy

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.

Article Details

How to Cite
Hue, N. N., & Huy, D. Q. (2015). Monotonicity of sequences involving generalized convexity function and sequences. Tamkang Journal of Mathematics, 46(2), 121–127. https://doi.org/10.5556/j.tkjm.46.2015.1626
Section
Papers
Author Biographies

Nguyen Ngoc Hue

Department of Natural Science and Technology, Tay Nguyen University, Daklak, Vietnam.

Duong Quoc Huy

Department of Natural Science and Technology, Tay Nguyen University, Daklak, Vietnam.

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