Geometric convexity of the generalized sine and the generalized hyperbolic sine

摘要

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}".