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

Year 2017, Volume: 5 Issue: 1, 77 - 84, 01.04.2017

Ladislav Matejıcka

https://izlik.org/JA98PH58UU

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.

Keywords

Convex combinations bounds; centroidal mean; harmonic mean; weighted geometric mean; logarithmic mean; identric mean.

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