Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means

Abstract

The logarithmic mean \(L(a,b)\), identric mean \(I(a,b)\), arithmeticmean \(A(a,b)\) and harmonic mean \(H(a,b)\) of two positive real values \(a\) and \(b\) are defined by\begin{align*}\label{main}&L(a,b)=\begin{cases}\tfrac{b-a}{\log b-\log a},& a\neq b,\\a,&a=b,\end{cases}\\&I(a,b)=\begin{cases}\tfrac{1}{{\rm e}}\left(\tfrac{b^b}{a^a}\right)^{\tfrac{1}{b-a}},& a\neq b,\\a,&a=b,\end{cases}\end{align*}\(A(a,b)=\tfrac{a+b}{2}\) and \(H(a,b)=\tfrac{2ab}{a+b}\), respectively. In this article, we answer the questions: What are the best possible parameters \(\alpha_{1},\alpha_{2},\beta_{1}\) and \(\beta_{2}\), such that \(\alpha_{1}A(a,b)+(1-\alpha_{1})H(a,b)\leq L(a,b)\leq\beta_{1}A(a,b)+(1-\beta_{1})H(a,b)\) and \(\alpha_{2}A(a,b)+(1-\alpha_{2})H(a,b)\leq I(a,b)\leq\beta_2A(a,b)+(1-\beta_{2})H(a,b)\) hold for all \(a,b>0\)?