OPTIMAL WEIGHTED GEOMETRIC MEAN BOUNDS OF CENTROIDAL AND HARMONIC MEANS FOR CONVEX COMBINATIONS OF LOGARITHMIC AND IDENTRIC MEANS

Abstract

In this paper, optimal weighted geometric mean bounds of centroidal and harmonic means for convex combination of logarithmic and identric means are proved. We find the greatest value $\gamma(\alpha)$ and the least value $\beta(\alpha)$ for each $\alpha\in (0,1)$ such that the double inequality: $C^{\gamma(\alpha)}(a,b)H^{1-\gamma(\alpha)}(a,b)<\alpha L(a,b)+({1-\alpha})I(a,b)<C^{\beta(\alpha)}(a,b)H^{1-\beta(\alpha)}(a,b)$ holds for all $a,b>0$ with $a\neq b.$ Here, $C(a,b),$ $H(a,b)$, $L(a,b),$ and $I(a,b)$ denote centroidal, harmonic, logarithmic and identric means of two positive numbers $a$ and $b,$ respectively.