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\begin{equation*}J_{p}(a,b)=\begin{cases}\tfrac{p(a^{p+1}-b^{p+1})}{(p+1)(a^p-b^p)}, & a\neq b,p\neq 0,-1,\\\tfrac{a-b}{\log{a}-\log{b}}, & a\neq b,p=0,\\\tfrac{ab(\log{a}-\log{b})}{a-b}, & a\neq b,p=-1,\\a, & a=b,\end{cases}\end{equation*}\(A(a,b)=\tfrac{a+b}{2}\), and \(H(a,b)=\tfrac{2ab}{a+b}\),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 \(a\neq b\)?