Geometric convexity of the generalized sine and the generalized hyperbolic sine

TL;DR

This paper proves geometric concavity of the generalized sine function and geometric convexity of the generalized hyperbolic sine function, confirming a conjecture related to their properties.

Contribution

It establishes the geometric convexity and concavity of generalized trigonometric functions, verifying a previously posed conjecture.

Findings

Generalized sine function is geometrically concave.

Generalized hyperbolic sine function is geometrically convex.

Confirmed conjecture from prior research.

Abstract

In the paper, the authors prove that the generalized sine function $\sin_{p,q}(x)$ and the generalized hyperbolic sine function $\sinh_{p,q}(x)$ are geometrically concave and geometrically convex, respectively. Consequently, the authors verify a conjecture posed in the paper "B. A. Bhayo and M. Vuorinen, On generalized trigonometric functions with two parameters, J. Approx. Theory 164 (2012), no.~10, 1415\nobreakdash--1426; Available online at \url{http://dx.doi.org/10.1016/j.jat.2012.06.003}".