Sharp Inequalities between Harmonic, Seiffert, Quadratic and Contraharmonic Means

TL;DR

This paper establishes sharp inequalities involving harmonic, Seiffert, quadratic, and contraharmonic means, identifying the exact bounds for these inequalities across all positive distinct real numbers.

Contribution

It determines the precise greatest and least values for parameters ensuring inequalities between various classical means hold universally.

Findings

Identified exact bounds for inequalities between means.

Provided sharp inequalities applicable to all positive distinct numbers.

Enhanced understanding of relationships among classical means.

Abstract

In this paper, we present the greatest values $\alpha$, $\lambda$ and $p$, and the least values $\beta$, $\mu$ and $q$ such that the double inequalities $\alpha D(a,b)+(1-\alpha)H(a,b)<T(a,b)<\beta D(a,b)+(1-\beta) H(a,b)$, $\lambda D(a,b)+(1-\lambda)H(a,b)<C(a,b)<\mu D(a,b)+(1-\mu) H(a,b)$ and $p D(a,b)+(1-p)H(a,b)<Q(a,b)<q D(a,b)+(1-q)H(a,b)$ hold for all $a,b>0$ with $a\neq b$, where $H(a,b)=2ab/(a+b)$, $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$, $Q(a,b)=\sqrt{(a^2+b^2)/2}$, $C(a,b)=(a^2+b^2)/(a+b)$ and $D(a,b)=(a^3+b^3)/(a^2+b^2)$ are the harmonic, Seiffert, quadratic, first contraharmonic and second contraharmonic means of $a$ and $b$, respectively.