Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means
We prove that the double inequalities Iα1(a,b)Q1-α1(a,b)<M(a,b)<Iβ1(a,b)Q1-β1(a,b),Iα2(a,b)C1-α2(a,b)<M(a,b)<Iβ2(a,b)C1-β2(a,b) hold for all a,b>0 with a≠b if and only if α1≥1/2, β1≤log[2log(1+2)]/(1-log2), α2≥5/7, and β2≤log[2log(1+2)], where I(a,b), M(a,b), Q(a,b), and C(a,b) are th...
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| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/348326 |
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| author | Tie-Hong Zhao Yu-Ming Chu Yun-Liang Jiang Yong-Min Li |
| author_facet | Tie-Hong Zhao Yu-Ming Chu Yun-Liang Jiang Yong-Min Li |
| author_sort | Tie-Hong Zhao |
| collection | DOAJ |
| description | We prove that the double inequalities Iα1(a,b)Q1-α1(a,b)<M(a,b)<Iβ1(a,b)Q1-β1(a,b),Iα2(a,b)C1-α2(a,b)<M(a,b)<Iβ2(a,b)C1-β2(a,b) hold for all a,b>0 with a≠b if and only if α1≥1/2, β1≤log[2log(1+2)]/(1-log2), α2≥5/7, and β2≤log[2log(1+2)], where I(a,b), M(a,b), Q(a,b), and C(a,b) are the identric, Neuman-Sándor, quadratic, and contraharmonic means of a and b, respectively. |
| format | Article |
| id | doaj-art-2cadabbfc97c4dfd88c1d50e2dd7afca |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-2cadabbfc97c4dfd88c1d50e2dd7afca2025-08-20T03:55:00ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/348326348326Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic MeansTie-Hong Zhao0Yu-Ming Chu1Yun-Liang Jiang2Yong-Min Li3Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, ChinaSchool of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, ChinaSchool of Information & Engineering, Huzhou Teachers College, Huzhou 313000, ChinaSchool of Automation, Southeast University, Nanjing 210096, ChinaWe prove that the double inequalities Iα1(a,b)Q1-α1(a,b)<M(a,b)<Iβ1(a,b)Q1-β1(a,b),Iα2(a,b)C1-α2(a,b)<M(a,b)<Iβ2(a,b)C1-β2(a,b) hold for all a,b>0 with a≠b if and only if α1≥1/2, β1≤log[2log(1+2)]/(1-log2), α2≥5/7, and β2≤log[2log(1+2)], where I(a,b), M(a,b), Q(a,b), and C(a,b) are the identric, Neuman-Sándor, quadratic, and contraharmonic means of a and b, respectively.http://dx.doi.org/10.1155/2013/348326 |
| spellingShingle | Tie-Hong Zhao Yu-Ming Chu Yun-Liang Jiang Yong-Min Li Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means Abstract and Applied Analysis |
| title | Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means |
| title_full | Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means |
| title_fullStr | Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means |
| title_full_unstemmed | Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means |
| title_short | Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means |
| title_sort | best possible bounds for neuman sandor mean by the identric quadratic and contraharmonic means |
| url | http://dx.doi.org/10.1155/2013/348326 |
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