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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:
 2.6e+61 Means