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


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


logarithmic mean, identric mean, arithmetic mean, harmonic mean


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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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