Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means

Abstract

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.