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

Authors : Ladislav MATEJICKA

Pages : 77-84

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Publication Date : 2017-04-01

Article Type : Research Paper

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 $\gammainsert ignore into journalissuearticles values(\alpha);$ and the least value $\betainsert ignore into journalissuearticles values(\alpha);$ for each $\alpha\in insert ignore into journalissuearticles values(0,1);$ such that the double inequality: $C^{\gammainsert ignore into journalissuearticles values(\alpha);}insert ignore into journalissuearticles values(a,b);H^{1-\gammainsert ignore into journalissuearticles values(\alpha);}insert ignore into journalissuearticles values(a,b);<\alpha Linsert ignore into journalissuearticles values(a,b);+insert ignore into journalissuearticles values({1-\alpha});Iinsert ignore into journalissuearticles values(a,b);0$ with $a\neq b.$ Here, $Cinsert ignore into journalissuearticles values(a,b);,$ $Hinsert ignore into journalissuearticles values(a,b);$, $Linsert ignore into journalissuearticles values(a,b);,$ and $Iinsert ignore into journalissuearticles values(a,b);$ 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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