A note on generalized trigonometric and hyperbolic functions

In this paper, we prove that the function x → log (x/sinp(x))/ log (sinhp(x)/x) (p ∈ [2,∞)) is strictly increasing on (0,πp/2) , where πp/2 = ∫ 1 0 (1− t p)−1/pdt , and sinp(x) and sinhp(x) denote the generalized trigonometric sine and generalized hyperbolic sine functions, respectively. As application, a conjecture due to Klen, Vuorinen and Zhang [J. Math. Anal. Appl. 409 (2014), 521–529] is proved, and the best positive constants α and β such that ( sinhp(x) x )α < x sinp(x) < ( sinhp(x) x )β