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

## Authors

• Wei-feng Xia Huzhou Teachers College
• Chu Yu-Ming Huzhou Teachers College

## Keywords:

logarithmic mean, identric mean, arithmetic mean, harmonic mean

## 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 $$a,b>0$$?

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2010-08-01

## How to Cite

Xia, W.- feng, & Yu-Ming, C. (2010). Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means. Rev. Anal. Numér. Théor. Approx., 39(2), 176–183. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2010-vol39-no2-art10

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