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

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.

References

• [1] Alzer, H. and Qiu, S. L., Inequalities for means in two variables, Arch. Math. (Basel), 80,(2003), no. 2, 201-215.
• [2] Bullen, P. S., Mitrinovic, D. S. and Vasic, P. M., Means and their inequalities, D. Reidel Publishing Co., Dordrecht, 1958.
• [3] Carlson, B. C., The logarithmic mean, Amer. Math. Monthly, 79,(1972), 615-618.
• [4] Chu, Y. M., Hou, S. W. and Xia, W.F., Optimal convex combinations bounds of centroidaland harmonic means for logarithmic and identric means, Buletin of the Iranian Mathematical Society, Vol. 39,(2013), no. 2, 259-269.
• [5] Kahling, P. and Matkowski, J., Functional equations involving the logarithmic mean, Z. Angew Math. Mech. 76,(1996), no. 7, 385-390.
• [6] Matejicka, L., Proof of One Optimal Inequalities for Generalized Logarithmic Arithmetic and Geometric Means, J. Inequal. Appl.,(2010), Article ID 902432, 5 pages.
• [7] Matejicka, L., Optimal convex combinations bounds of centroidal and harmonic means for weighted geometric mean of logarithmic and identric means, Journal of mathematical in equalities,(2014), Volume 8, no. 4, 939-945.
• [8] Pitinger, A. O., The logarithmic mean in n variables, Amer. Math. Monthly 92,(1985), no. 2, 99-104.
• [9] Polya, G. and Szeg}o, G., Isoperimetric inequalities in mathematical physics, Princeton University Press, Princeton, 1951.
• [10] Vavro, J., Kopecky, M. and Vavro, J. jr., Nove prostriedky a metody riesenia sustav telies III- 1.vyd., Zilina, 2007, ISBN 978-80-8075-256-9.
• [11] Seiffert, W., Problem 887, Nieuw Archief voor Wiskunde, Vol. 11, no.2, 176-176.
• [12] Shaoqin, G., Hongya, G. and Wenying, S., Optimal convex combination bounds of the centroidal and harmonic means for the seiert mean, International Journal of Pure and AppliedMathematics, Volume 70,(2011), no. 5, 701-709.
• [13] Yang, Z.-H., New sharp bounds for logarithmic mean and identric mean, Journal of Inequalities and Applications, (2013), 116, 17 pages.