Inequalities related to the arithmetic, geometric and harmonic means

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

A Note on the Weighted Harmonic–geometric–arithmetic Means Inequalities

In this note, we derive non trivial sharp bounds related to the weighted harmonicgeometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of a symmetric positive definite matrix and an inequality related to the coefficients of polynomials with positive roots. Mathematics subject classification (201...

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 for the power, harmonic and logarithmic means

For all $a,b>0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers...

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...