Mixed arithmetic and geometric means

Mixed Arithmetic and Geometric Means and Related Inequalities

Mixed arithmetic and geometric means, with and without weights, are both considered. Related to mixed arithmetic and geometric means, the following three types of inequalities and their generalizations, from three variables to a general n variables, are studied. For arbitrary x, y, z ≥ 0 we have [ x + y + z 3 (xyz) ]1/2 ≤ ( x + y 2 · y + z 2 · z + x 2 )1/3 , (A) 1 3 (√ xy + √ yz + √ zx ) ≤ 1 2 ...

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

Concentration of the Ratio between the Geometric and Arithmetic Means

This paper is motivated by the article [GluMi] by E. Gluskin and V. Milman, who considered the variant ∏n i=1 |yi| ≤ √ n−1 ∑n i=1 y 2 i of the AM-GM inequality in the equal weights case. Roughly speaking, they showed that the ratio ∏n i=1 |yi|/ √ n−1 ∑n i=1 y 2 i is bounded below by 0.394 asymptotically in n and with high probability, where probability refers to Haar measure on the euclidean un...

An Inequality between Compositions of Weighted Arithmetic and Geometric Means

Let P denote the collection of positive sequences defined on N. Fix w ∈ P. Let s, t, respectively, be the sequences of partial sums of the infinite series ∑ wk and ∑ sk, respectively. Given x ∈ P, define the sequences A(x) and G(x) of weighted arithmetic and geometric means of x by An(x) = n ∑ k=1 wk sn xk, Gn(x) = n ∏ k=1 x wk/sn k , n = 1, 2, . . . Under the assumption that log t is concave, ...

Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means

Copyright q 2010 B.-Y. Long and Y.-M. Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For p ∈ R, the generalized logarithmic mean L p a, b, arithmetic mean Aa, b, and geometric mean Ga, b of two positive numbers a and b are d...