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Inequalities for generalized trigonometric and hyperbolic sine functions

Authors
  • Wang, Miao-Kun
  • Chu, Yu-Ming
  • Jiang, Yue-Ping
Type
Preprint
Publication Date
Dec 05, 2012
Submission Date
Dec 05, 2012
Identifiers
arXiv ID: 1212.4681
Source
arXiv
License
Yellow
External links

Abstract

We prove that the inequalities $\sin_{p,q}(\sqrt{rs})\geq \sqrt{\sin_{p,q}(r)\sin_{p,q}(s)}$ and $\sinh_{p,q}(\sqrt{r^*s^*}) \leq \sqrt{\sinh_{p,q}(r^*)\sinh_{p,q}(s^*)}$ hold for all $p,q\in(1,\infty)$, $r,s\in(0,\int_{0}^{1}(1-t^q)^{-1/p}dt)$ and $r^*,s^*\in(0,\int_{0}^{\infty}(1+t^q)^{-1/p}dt)$, 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)].

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