Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean

Abstract

Abstract In the article, we prove that λ1=1/2+[(2+log(1+2))/2]1/ν−1/2 $\lambda _{1}=1/2+\sqrt{ [ (\sqrt{2}+ \log (1+\sqrt{2}) )/2 ]^{1/\nu }-1}/2$, μ1=1/2+6ν/(12ν) $\mu _{1}=1/2+\sqrt{6 \nu }/(12\nu )$, λ2=1/2+[(π+2)/4]1/ν−1/2 $\lambda _{2}=1/2+\sqrt{ [(\pi +2)/4 ] ^{1/\nu }-1}/2$ and μ2=1/2+3ν/(6ν) $\mu _{2}=1/2+\sqrt{3\nu }/(6\nu )$ are the best possible parameters on the interval [1/2,1] $[1/2, 1]$ such that the double inequalities Cν[λ1x+(1−λ1)y,λ1y+(1−λ1)x]A1−ν(x,y)0 $x, y>0$ with x≠y $x\neq y$ and ν∈[1/2,∞) $\nu \in [1/2, \infty )$, where A(x,y) $A(x, y)$ is the arithmetic mean, C(x,y) $C(x, y)$ is the contraharmonic mean, and RQA(x,y) $\mathcal{R}_{QA}(x, y)$ and RAQ(x,y) $\mathcal{R}_{AQ}(x, y)$ are two Neuman means.

Keywords

• Arithmetic mean
• Quadratic mean
• Contraharmonic mean
• Schwab–Borchardt mean
• Neuman mean
• Two-parameter contraharmonic and arithmetic mean