An optimal double inequality among the one-parameter, arithmetic and harmonic means

Abstract

For $p\in \mathbb{R}$, the one-parameter mean $J_p(a,b)$, arithmetic mean $A(a,b)$, and harmonic mean $H(a,b)$ of two positive real numbers $a$ and $b$ are defined by$$J_p(a,b)=\{\begin{array}{ll}\frac{p(a^{p+1}-b^{p+1})}{(p+1)(a^p-b^p)},& a\ne b,p\ne 0,-1,\\ \frac{a-b}{\log a-\log b},& a\ne b,p=0,\\ \frac{ab(\log a-\log b)}{a-b},& a\ne b,p=-1,\\ a,& a=b,\end{array}$$$A(a,b)=\frac{a+b}{2}$, and $H(a,b)=\frac{2ab}{a+b}$,respectively. In this paper, we answer the question: For $\alpha \in (0,1)$, what are the greatest value $r_1$ and the least value $r_2$ such that the double inequality $J_{r_1}(a,b)<\alpha A(a,b)+(1-\alpha )H(a,b)<J_{r_2}(a,b)$ holds for all $a,b>0$ with $a\ne b$?