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The optimal geometric combination bounds for Neuman means of harmonic, arithmetic and contra-harmonic. (English) Zbl 1326.26057
Summary: In the paper, we find the greatest values \(\lambda_1\), \(\lambda_2\), \(\lambda_3\), \(\lambda_4\) and the least values \(\mu_1\), \(\mu_2\), \(\mu_3\), \(\mu_4\) such that the double inequalities
\[ A^{\lambda_1}(a,b)H^{1-\lambda_1}(a,b)<N_{AG}(a,b)< A^{\mu_1}(a,b)H^{1-\mu_1}(a,b), \]
\[ C^{\lambda_2}(a,b)A^{1-\lambda_2}(a,b)<N_{QA}(a,b)< C^{\mu_2}(a,b)A^{1-\mu_2}(a,b), \]
\[ A^{\lambda_3}(a,b)H^{1-\lambda_3}(a,b)<N_{GA}(a,b)<A^{\mu_3}(a,b)H^{1-\mu_3}(a,b), \]
\[ C^{\lambda_4}(a,b)A^{1-\lambda_4}(a,b)<N_{AQ}(a,b)<C^{\mu_4}(a,b)A^{1-\mu_4}(a,b) \]
hold for all \(a,b>0\) with \(a\neq b\). Here \(H(a,b)\), \(G(a,b)\), \(A(a,b)\), \(Q(a,b)\), \(C(a,b)\) respectively denote the harmonic, geometric, arithmetic, quadratic and contra-harmonic of \(a\) and \(b\), and \(N_{AG}(a,b)\), \(N_{GA}(a,b)\), \(N_{QA}(a,b)\) and \(N_{AQ}(a,b)\) are four Neuman means derived from the Schwab-Borchardt mean.
MSC:
26E60 Means
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