Optimal Convex Combinations Bounds of Centroidal and Harmonic Means for Weighted Geometric Mean of Logarithmic and Identric 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)

Integral Representations for Bivariate Complex Geometric Mean and Applications

In the paper, the authors survey integral representations (including the Lévy– Khintchine representations) and applications of some bivariate means (including the logarithmic mean, the identric mean, Stolarsky’s mean, the harmonic mean, the (weighted) geometric means and their reciprocals, and the Toader–Qi mean) and the multivariate (weighted) geometric means and their reciprocals, derive inte...

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 bounds for Neuman means in terms of geometric, arithmetic and quadratic means

*Correspondence: [email protected] 2School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of the article Abstract In this paper, we present sharp bounds for the two Neuman means SHA and SCA derived from the Schwab-Borchardt mean in terms of convex combinations of either the weighted arithmetic and ...

Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means