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

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

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...

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...

Best Possible Inequalities among Harmonic, Geometric, Logarithmic and Seiffert Means

In this paper, we find the greatest value α and the least values β , p , q and r in (0,1/2) such that the inequalities L(αa+ (1−α)b,αb+ (1−α)a) < P(a,b) < L(βa + (1− β)b,βb + (1− β)a) , H(pa + (1− p)b, pb + (1− p)a) > G(a,b) , H(qa+ (1− q)b,qb +(1− q)a) > L(a,b) , and G(ra+(1− r)b,rb+(1− r)a) > L(a,b) hold for all a,b > 0 with a = b . Here, H(a,b) , G(a,b) , L(a,b) and P(a,b) denote the harmoni...

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...