×

Sharp inequalities between means. (English) Zbl 1223.26049

Summary: For \(p\in\mathbb R\), the \(p\)-th power mean \(M_p(a,b)\), arithmetic mean \(A(a,b)\), geometric mean \(G(a,b)\), and harmonic mean \(H(a,b)\) of two positive numbers \(a\) and \(b\) are defined by
\[ M_p(a,b)=\begin{cases}\left(\frac{a^p+b^p}{w}\right),\quad p\neq 0,\\ \sqrt{ab},\quad p=0\end{cases}, \]
\(A(a,b)=(a+b)/2\), \(G(a,b)=\sqrt{ab}\), and \(H(a,b)=2ab/(a+b)\), respectively.
In this paper, we answer the questions: For \(\alpha\in(0,1)\), what are the greatest values \(p\), \(r\) and \(m\), and the least values \(q\), \(s\) and \(n\), such that the inequalities \(M_p(a,b)\leq A^\alpha(a,b)G^{1-\alpha}(a,b)\leq M_q(a,b)\), \(M_r(a,b)\leq G^\alpha(a,b)H^{1-\alpha}(a,b)\leq M_s(a,b)\) and \(M_m(a,b)\leq A^\alpha(a,b)H^{1-\alpha}(a,b)\leq M_n(a,b)\) hold for all \(a,b>0\)?

MSC:

26E60 Means
PDFBibTeX XMLCite
Full Text: DOI