Exact inequalities involving power mean, arithmetic mean and identric mean

Abstract

For \(p\in \mathbb{R}\), the power mean \(M_{p}(a,b)\) of order \(p\), identric mean \(I(a,b)\) and arithmetic mean \(A(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by \begin{equation*} M_{p}(a,b)= \begin{cases} \displaystyle\left(\tfrac{a^{p}+b^{p}}{2}\right)^{1/p}, & p\neq 0,\\ \sqrt{ab}, & p=0, \end{cases} \quad I(a,b)= \begin{cases} \displaystyle\tfrac{1}{\rm {e}}\left(b^{b}/a^{a}\right)^{1/(b-a)}, & a\neq b,\\ \displaystyle a, & a=b, \end{cases} \end{equation*} and \(A(a,b)=(a+b)/2\), respectively. In the article, we answer the questions: What are the least values \(p\), \(q\) and \(r\), such that inequalities \(A^{1/2}(a,b)I^{1/2}(a,b)\leq M_{p}(a,b)\), \(A(a,b)^{1/3}I^{2/3}(a,b)\leq M_{q}(a,b)\) and \(A^{2/3}(a,b)I^{1/3}(a,b)\leq M_{r}(a,b)\) hold for all \(a,b>0\)?

Keywords

• power mean
• identric mean
• arithmetic mean