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

## Authors

• Wang Miao-Kun Huzhou Teachers College
• Qiu Ye-Fang Huzhou Teachers College
• Chu Yu-Ming Huzhou Teachers College

## Keywords:

one-parameter mean, arithmetic mean, harmonic mean

## 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 $$a\neq b$$?

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2010-08-01

## How to Cite

Miao-Kun, W., Ye-Fang, Q., & Yu-Ming, C. (2010). An optimal double inequality among the one-parameter, arithmetic and harmonic means. Rev. Anal. Numér. Théor. Approx., 39(2), 169–175. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2010-vol39-no2-art9

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