Optimal Inequalities between Harmonic, Geometric, Logarithmic, and Arithmetic-Geometric Means

Abstract

We find the least values $p$, $q$, and $s$ in (0, 1/2) such that the inequalities $H(pa+(1-p)b$, $pb+(1-p)a)>\mathrm{AG}(a,b)$, $G(qa+(1-q)b,qb+(1-q\left)a\right)>\mathrm{AG}(a,b)$, and $L(sa+(1-s)b,sb+(1-s\left)a\right)>\mathrm{AG}(a,b)$ hold for all $a,b>0$ with $a\ne b$, respectively. Here $\mathrm{AG}(a,b),H(a,b),G(a,b)$, and $L(a,b)$ denote the arithmetic-geometric, harmonic, geometric, and logarithmic means of two positive numbers a and b, respectively.