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

Author:

Xia Wei-feng,Yu-Ming Chu

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 \(a,b>0\)?

Publisher

Academia Romana Filiala Cluj

Subject

Applied Mathematics,Computational Mathematics,Numerical Analysis,Mathematics (miscellaneous)

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Exact inequalities involving power mean, arithmetic mean and identric mean;Journal of Numerical Analysis and Approximation Theory;2011-08-01

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