Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means - Yu-Ming Chu; Miao-Kun Wang; Bao-Yu Liu (Journal of Numerical Analysis and Approximation Theory, 2013)
In this paper, we find the greatest values \(\alpha\) and \(\lambda\), and the least values \(\beta\) and \(\mu\) such that the double inequalities \[C^{\alpha}(a,b)A^{1-\alpha}(a,b)<M(a,b)<C^{\beta}(a,b)A^{1-\beta}(a,b)\] and \begin{align*} &[C(a,b)/6+5 A(a,b)/6]^{\lambda }\left[C^{1/6}(a,b)A^{5/6}(a,b)\right]^{1-\lambda}<M(a,b)<\\ &\qquad<[C(a,b)/6+5 A(a,b)/6]^{\mu}\left[C^{1/6}(a,b)A^{5/6}(a,b)\right]^{1-\mu} \end{align*} hold for all \(a,b>0\) with \(a\neq b\), where \(M(a,b)\), \(A(a,b)\) and \(C(a,b)\) denote the Neuman-Sándor, arithmetic, and contra-harmonic means of \(a\) and \(b\), respectively.