Sharp two parameter bounds for logarithmic and arithmetic-geometric means

TL;DR

This paper establishes sharp bounds for inequalities involving logarithmic and arithmetic-geometric means, providing precise conditions on parameters for these inequalities to hold universally for positive distinct numbers.

Contribution

It introduces exact parameter bounds for inequalities relating various classical means, enhancing understanding of their relationships and inequalities.

Findings

Derived necessary and sufficient conditions for inequalities to hold universally.

Established sharp bounds for parameters in mean inequalities.

Enhanced theoretical understanding of mean relationships.

Abstract

For fixed $s\geq 1$ and $t_{1},t_{2}\in(0,1/2)$ we prove that the inequalities $G^{s}(t_{1}a+(1-t_{1})b,t_{1}b+(1-t_{1})a)A^{1-s}(a,b)>AG(a,b)$ and $G^{s}(t_{2}a+(1-t_{2})b,t_{2}b+(1-t_{2})a)A^{1-s}(a,b)>L(a,b)$ hold for all $a,b>0$ with $a\neq b$ if and only if $t_{1}\geq 1/2-\sqrt{2s}/(4s)$ and $t_{2}\geq 1/2-\sqrt{6s}/(6s)$. Here $G(a,b)$, $L(a,b)$, $AG(a,b)$ and $A(a,b)$ are the geometric, logarithmic, arithmetic-geometric and arithmetic means of $a$ and $b$, respectively.