Optimal inequalities for the power, harmonic and logarithmic means

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 convex combinations bounds of centrodial and harmonic means for logarithmic and identric means

We find the greatest values $alpha_{1} $ and $alpha_{2} $, and the least values $beta_{1} $ and $beta_{2} $ such that the inequalities $alpha_{1} C(a,b)+(1-alpha_{1} )H(a,b)

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

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, ...

Optimal Inequalities for Power Means

optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means

we find the greatest values $alpha_{1} $ and $alpha_{2} $, and the least values $beta_{1} $ and $beta_{2} $ such that the inequalities $alpha_{1} c(a,b)+(1-alpha_{1} )h(a,b)