Generalized power means and interpolating inequalities

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...

Inequalities for Generalized Logarithmic Means

For p ∈ R, the generalized logarithmic mean Lp of two positive numbers a and b is defined as Lp a, b a, for a b, LP a, b b 1 − a 1 / p 1 b − a 1/p , for a/ b, p / − 1, p / 0, LP a, b b − a / log b − loga , for a/ b, p −1, and LP a, b 1/e b/a 1/ b−a , for a/ b, p 0. In this paper, we prove that G a, b H a, b 2L−7/2 a, b , A a, b H a, b 2L−2 a, b , and L−5 a, b H a, b for all a, b > 0, and the co...

Optimal Inequalities for Power Means

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...