Parameter convexity and concavity of generalized trigonometric functions

Abstract

We study the convexity properties of the generalized trigonometric functions considered as functions of parameter. We show that $p\to {\sin }_p(y)$ and $p\to {\cos }_p(y)$ are log-concave on the appropriate intervals while $p\to {\tan }_p(y)$ is log-convex. We also prove similar facts about the generalized hyperbolic functions. In particular, our results settle the major part of a conjecture put forward in a recent paper by Baricz, Bhayo and Vuorinen.

15 pages, no figures

References in corpus (2)

• Turán type inequalities for generalized inverse trigonometric functions
• Inequalities for generalized trigonometric and hyperbolic sine functions