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Best possible bounds for Neuman-Sándor mean by the identric, quadratic and contraharmonic means. (English) Zbl 1276.26065

Summary: We prove that the double inequalities \(I^{\alpha_1}(a, b)Q^{1 - \alpha_1}(a, b) < M(a, b) < I^{\beta_1}(a, b)Q^{1 - \beta_1}(a, b)\), \(I^{\alpha_2}(a, b)C^{1 - \alpha_2}(a, b) < M(a, b) < I^{\beta_2}(a, b)C^{1 - \beta_2}(a, b)\) hold for all \(a, b > 0\) with \(a \neq b\) if and only if \(\alpha_1 \geq 1/2\), \(\beta_1 \leq \log[\sqrt{2}\log(1 + \sqrt{2})]/(1 - \log \sqrt{2})\), \(\alpha_2 \geq 5/7\), and \(\beta_2 \leq \log[2\log(1 + \sqrt{2})]\), where \(I(a, b)\), \(M(a, b)\), \(Q(a, b)\), and \(C(a, b)\) are the identric, Neuman-Sándor, quadratic, and contraharmonic means of \(a\) and \(b\), respectively.

MSC:

26E60 Means
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