Geometric convexity of the generalized sine and the generalized hyperbolic sine

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}”.