Parameter convexity and concavity of generalized trigonometric functions

TL;DR

This paper investigates the convexity and concavity properties of generalized trigonometric and hyperbolic functions as functions of their parameters, establishing log-concavity and log-convexity results that resolve a recent conjecture.

Contribution

It provides a comprehensive analysis of the parameter convexity properties of generalized trigonometric functions, settling a significant conjecture in the field.

Findings

$p o ext{sin}_p(y)$ and $p o ext{cos}_p(y)$ are log-concave

$p o ext{tan}_p(y)$ is log-convex

Results apply to generalized hyperbolic 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.