Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means
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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Publishing House of the Romanian Academy
2013-08-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
| Subjects: | |
| Online Access: | https://www.ictp.acad.ro/jnaat/journal/article/view/987 |
| Summary: | 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. |
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| ISSN: | 2457-6794 2501-059X |