Convexity properties of generalized trigonometric and hyperbolic functions

TL;DR

This paper investigates the convexity properties of generalized trigonometric and hyperbolic functions, focusing on their power mean inequalities with two parameters, expanding understanding of these functions introduced by Lindqvist and Takeuchi.

Contribution

It provides new insights into the convexity and power mean inequalities of generalized trigonometric and hyperbolic functions with two parameters.

Findings

Established convexity properties of generalized functions

Derived power mean inequalities for these functions

Extended previous results by Lindqvist and Takeuchi

Abstract

We study the power mean inequality of the generalized trigonometric and hyperbolic functions with two parameters. The generalized $p$-trigonometric and $(p, q)$-trigonometric functions were introduced by P. Lindqvist and S. Takeuchi, respectively.