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://ictp.acad.ro/jnaat/journal/article/view/987 |
| _version_ | 1818529314552938496 |
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| author | Yu-Ming Chu Miao-Kun Wang Bao-Yu Liu |
| author_facet | Yu-Ming Chu Miao-Kun Wang Bao-Yu Liu |
| author_sort | Yu-Ming Chu |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-11T17:05:18Z |
| format | Article |
| id | doaj.art-e68e188705344ca9979f8a0c3494b255 |
| institution | Directory Open Access Journal |
| issn | 2457-6794 2501-059X |
| language | English |
| last_indexed | 2024-12-11T17:05:18Z |
| publishDate | 2013-08-01 |
| publisher | Publishing House of the Romanian Academy |
| record_format | Article |
| series | Journal of Numerical Analysis and Approximation Theory |
| spelling | doaj.art-e68e188705344ca9979f8a0c3494b2552022-12-22T00:57:43ZengPublishing House of the Romanian AcademyJournal of Numerical Analysis and Approximation Theory2457-67942501-059X2013-08-01422Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic meansYu-Ming Chu0Miao-Kun Wang1Bao-Yu Liu2Huzhou Teachers CollegeHuzhou Teachers CollegeHangzhou Dianzi UniversityIn 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.https://ictp.acad.ro/jnaat/journal/article/view/987Neuman-Sándor meanarithmetic meancontra-harmonic mean |
| spellingShingle | Yu-Ming Chu Miao-Kun Wang Bao-Yu Liu Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means Journal of Numerical Analysis and Approximation Theory Neuman-Sándor mean arithmetic mean contra-harmonic mean |
| title | Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means |
| title_full | Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means |
| title_fullStr | Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means |
| title_full_unstemmed | Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means |
| title_short | Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means |
| title_sort | sharp inequalities for the neuman sandor mean in terms of arithmetic and contra harmonic means |
| topic | Neuman-Sándor mean arithmetic mean contra-harmonic mean |
| url | https://ictp.acad.ro/jnaat/journal/article/view/987 |
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