Inequalities for Means in Two Variables

Abstract

We present various new inequalities involving the logarithmic mean L(x, y) = (x − y)/(log x − log y), the identric mean I (x, y) = (1/e)(x x /y y ) 1/(x−y) , and the classical arithmetic and geometric means, A(x, y) = (x + y)/2 and G(x, y) = √ xy. In particular, we prove the following conjecture, which was published in 1986 in this journal. If M r (x, y) = (x r /2+y r /2) 1/r (r = 0) denotes the power mean of order r, thenwith the best possible parameter c = (log 2)/(1 + log 2).

Refinements for mean-inequalities via the stabilizability concept

“…In the past years, enormous efforts by some authors has been devoted to refine various inequalities between means (called mean-inequalities), see [2][3][4][5][6][7][8][9][10] for instance and the related references cited therein. Our fundamental goal in this article is to explore the stabilizability concept for obtaining a game of mean-inequalities whose certain of them have been differently discussed in the literature.…”

The best possible Lehmer mean bounds for a convex combination of logarithmic and harmonic means

“…Alzer and Qiu [30] proved that M c (a, b) < The purpose of this paper is to answer the question: For α ∈ (0, 1), what are the largest value p and least value q such that the double inequality…”

“…In particular, many remarkable inequalities for the logarithmic mean can be found in literatures [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. It might be surprising that the logarithmic mean has applications in physics [31], economics [32], and even in meteorology [33].…”

Sharp two-parameter bounds for the identric mean

“…Many articles studying the properties of means of two variables have been published, and there is a large body of mathematical literature about comparing pairs of means. The interested reader may consult [1–3, 5–7, 9–11] and the references therein.…”