[Paper Review] Inequalities for generalized trigonometric and hyperbolic sine functions

Miao-Kun Wang, Yu-Ming Chu+1 more|arXiv (Cornell University)|Dec 5, 2012

Mathematical Inequalities and Applications5 references6 citations

TL;DR

This paper proves two sharp inequalities for generalized trigonometric and hyperbolic sine functions: $σ_{p,q}(\sqrt{rs}) \geq \sqrt{\sigma_{p,q}(r)\sigma_{p,q}(s)}$ and $\sinh_{p,q}(\sqrt{r^*s^*}) \leq \sqrt{\sinh_{p,q}(r^*)\sinh_{p,q}(s^*)}$ for $p,q > 1$ and appropriate domains. The authors establish these via logarithmic differentiation and monotonicity analysis of inverse functions, resolving a conjecture by Bhayo and Vuorinen and extending classical inequalities to the generalized $(p,q)$-framework.

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)].

Motivation & Objective

• To resolve a conjecture by Bhayo and Vuorinen (2012) on functional inequalities involving generalized trigonometric and hyperbolic sine functions.
• To establish sharp inequalities for $\sin_{p,q}$ and $\sinh_{p,q}$ using the geometric mean of arguments.
• To extend classical inequalities such as Mitrinović-Adamović and Lazarević to the generalized $(p,q)$-function framework.
• To analyze the monotonicity properties of inverse functions $\arcsin_{p,q}$ and $\mathrm{arcsinh}_{p,q}$ via differential inequalities.
• To provide a unified treatment of generalized sine and hyperbolic sine functions using integral representations and special function theory.

Proposed method

• Derive a lower bound for $\arcsin_{p,q}(x)$ using a rational function, proving it strictly exceeds the bound via derivative analysis.
• Establish an upper bound for $\frac{x}{\mathrm{arcsinh}_{p,q}(x)}$ using a rational expression, validated through case analysis on $p$ and $q$.
• Define the logarithmic mean-type function $H_{p_1}(x,y)$ and analyze its monotonicity using logarithmic differentiation of the ratio $J(x,y)$.
• Apply logarithmic differentiation to the ratio $J^*(x,y)$ involving $\mathrm{arcsinh}_{p,q}$ to analyze monotonicity in the hyperbolic case.
• Use the monotonicity of $F(x)$ and $F^*(x)$ derived from logarithmic derivatives to determine conditions under which $J$ and $J^*$ are increasing or decreasing.
• Apply the results to the inverse functions $\sin_{p,q}$ and $\sinh_{p,q}$ by substituting $x = \sin_{p,q}(r)$, $y = \sin_{p,q}(s)$, and $x = \sinh_{p,q}(r^*)$, $y = \sinh_{p,q}(s^*)$ respectively.

Experimental results

Research questions

• RQ1 Does the inequality $\sin_{p,q}(\sqrt{rs}) \geq \sqrt{\sin_{p,q}(r)\sin_{p,q}(s)}$ hold for all $p,q > 1$ and $r,s \in (0, \pi_{p,q}/2)$?
• RQ2 Does the inequality $\sinh_{p,q}(\sqrt{r^*s^*}) \leq \sqrt{\sinh_{p,q}(r^*)\sinh_{p,q}(s^*)}$ hold for all $p,q > 1$ and $r^*,s^* \in (0, m_{p,q}^*)$?
• RQ3 Can the conjecture by Bhayo and Vuorinen on generalized sine and hyperbolic sine inequalities be confirmed using integral and monotonicity methods?
• RQ4 What conditions on the power mean parameter $p_1$ or $p_2$ ensure the monotonicity of the ratio functions $J(x,y)$ and $J^*(x,y)$?
• RQ5 How do the asymptotic behaviors of $\arcsin_{p,q}(x)$ and $\mathrm{arcsinh}_{p,q}(x)$ at $x \to 0$ and $x \to \infty$ influence the derived inequalities?

Key findings

• The inequality $\sin_{p,q}(\sqrt{rs}) \geq \sqrt{\sin_{p,q}(r)\sin_{p,q}(s)}$ holds for all $p,q > 1$ and $r,s \in (0, \pi_{p,q}/2)$, with equality if and only if $r = s$.
• The inequality $\sinh_{p,q}(\sqrt{r^*s^*}) \leq \sqrt{\sinh_{p,q}(r^*)\sinh_{p,q}(s^*)}$ holds for all $p,q > 1$ and $r^*,s^* \in (0, m_{p,q}^*)$, with equality if and only if $r^* = s^*$.
• The authors prove a sharp lower bound for $\arcsin_{p,q}(x)$: $\arcsin_{p,q}(x) > \frac{px(1-x^q)^{1-1/p}}{(q-p)x^q + p}$ for $x \in (0,1)$.
• A sharp upper bound for $\frac{x}{\mathrm{arcsinh}_{p,q}(x)}$ is established: $\frac{x}{\mathrm{arcsinh}_{p,q}(x)} > \frac{(p-q)x^q + p}{p(1+x^q)^{1-1/p}}$ for $x \in (0,\infty)$.
• The integral $m_{p,q}^* = \int_0^\infty (1+t^q)^{-1/p} dt > 1$ is proven, ensuring the domain of $\sinh_{p,q}$ is sufficiently large for the inequalities to be meaningful.
• The proof technique via logarithmic differentiation and monotonicity of $F(x)$ and $F^*(x)$ confirms the conjecture by Bhayo and Vuorinen and extends it to the full parameter range $p,q > 1$.