Self-improvement of the inequality between arithmetic and geometric means

A Refinement of the Inequality between Arithmetic and Geometric Means

In this note we present a refinement of the AM-GM inequality, and then we estimate in a special case the typical size of the improvement. (1) exp 2 1 − n i=1 α i x

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, ...

the relationship between academic self-concept and academic achievement in english and general subjects of the students of high school

according to research, academic self-concept and academic achievement are mutually interdependent. in the present study, the aim was to determine the relationship between the academic self-concept and the academic achievement of students in english as a foreign language and general subjects. the participants were 320 students studying in 4th grade of high school in three cities of noor, nowshah...

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...

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