Sharp two parameter bounds for logarithmic and arithmetic-geometric means

Abstract

For fixed $s\ge 1$ 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 $a\ne b$ if and only if $t_1\ge 1/2-\sqrt{2s}/(4s)$ and $t_2\ge 1/2-\sqrt{6s}/(6s)$. 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.

8 pages