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Sharp power-type Heronian mean bounds for the Sándor and Yang means. (English) Zbl 1372.26032
Summary: We prove that the double inequalities \(H_{\alpha}(a, b) < X(a, b)<H_{\beta}(a, b)\) and \(H_{\lambda}(a, b)< U(a, b)<H_{\mu}(a, b)\) hold for all \(a, b>0\) with \(a\neq b\) if and only if \(\alpha\leq 1/2\), \(\beta\geq\log 3/(1+\log2)=0.6488\dots\), \(\lambda\leq 2\log 3/(2\log\pi-\log 2) =1.3764\dots\), and \(\mu\geq 2\), where \(H_{p}(a, b)\), \(X(a, b)\), and \(U(a, b)\) are, respectively, the \(p\)th power-type Heronian mean, Sándor mean, and Yang mean of \(a\) and \(b\).

26E60 Means
Full Text: DOI
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