Inequalities for generalized trigonometric and hyperbolic sine functions

Miao-Kun Wang, Yu-Ming Chu, Yue-Ping Jiang

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…