Inequalities for generalized trigonometric and hyperbolic sine functions

Abstract

We prove that the inequalities ${\sin }_{p,q}(\sqrt{rs})\ge \sqrt{{\sin }_{p,q}(r){\sin }_{p,q}(s)}$ and ${\sinh }_{p,q}(\sqrt{r^{\ast }{s}^{\ast }})\le \sqrt{{\sinh }_{p,q}(r^{\ast }){\sinh }_{p,q}(s^{\ast })}$ hold for all $p,q\in (1,\infty )$, $r,s\in (0,{\int }_0^1(1-t^q{)}^{-1/p}dt)$ and $r^{\ast },s^{\ast }\in (0,{\int }_0^{\infty }(1+t^q{)}^{-1/p}dt)$, where ${\sin }_{p,q}$ and ${\sinh }_{p,q}$ are the generalized trigonometric and hyperbolic sine functions, respectively. As a consequence of the results, we prove a conjecture due to Bhayo and Vuorinen [J. Approx. Theory, 164(2012)].

7 pages