On matrix inequalities between the power means: Counterexamples

Power matrix means and related inequalities ∗

This survey paper contains recent results for power matrix means and related inequalities for convex functions, Hadamard product of matrices as well as some inequalities involving exponential function of matrices.

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 Power Means

Optimal Inequalities between Seiffert’s Mean and Power Means

In this paper optimal inequalities between Seiffert’s mean and power means are derived using a simple monotony property. Mathematics subject classification (2000): 26E60, 26D05.

Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means

We prove that αH a, b 1 − α L a, b > M 1−4α /3 a, b for α ∈ 0, 1 and all a, b > 0 with a/ b if and only if α ∈ 1/4, 1 and αH a, b 1 − α L a, b < M 1−4α /3 a, b if and only if α ∈ 0, 3√345/80 − 11/16 , and the parameter 1 − 4α /3 is the best possible in either case. Here, H a, b 2ab/ a b , L a, b a − b / loga − log b , and Mp a, b a b /2 1/p p / 0 and M0 a, b √ ab are the harmonic, logarithmic, ...