Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means
For a,b>0 with a≠b, the Schwab-Borchardt mean SB(a,b) is defined as SB(a,b)={b2-a2/cos-1(a/b) if a<b,a2-b2/cosh-1(a/b) if a>b. In this paper, we find the greatest values of α1 and α2 and the least values of β1 and β2 in [0,1/2] such that H(α1a+(1-α1)b...
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2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/807623 |
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| author | Zai-Yin He Yu-Ming Chu Miao-Kun Wang |
| author_facet | Zai-Yin He Yu-Ming Chu Miao-Kun Wang |
| author_sort | Zai-Yin He |
| collection | DOAJ |
| description | For a,b>0 with a≠b, the Schwab-Borchardt mean SB(a,b) is defined as SB(a,b)={b2-a2/cos-1(a/b) if a<b,a2-b2/cosh-1(a/b) if a>b. In this paper, we find the greatest values of
α1 and α2 and the least values of
β1 and β2 in [0,1/2] such that H(α1a+(1-α1)b,α1b+(1-α1)a)<SAH(a,b)<H(β1a+(1-β1)b,β1b+(1-β1)a) and H(α2a+(1-α2)b,α2b+(1-α2)a)<SHA(a,b)<H(β2a+(1-β2)b,β2b+(1-β2)a). Similarly, we also find the greatest values of α3 and α4 and the least values of β3 and β4 in [1/2,1] such that C(α3a+(1-α3)b,α3b+(1-α3)a)<SCA(a,b)<C(β3a+(1-β3)b,β3b+(1-β3)a) and C(α4a+(1-α4)b,α4b+(1-α4)a)<SAC(a,b)<C(β4a+(1-β4)b,β4b+(1-β4)a). Here, H(a,b)=2ab/(a+b), A(a,b)=(a+b)/2, and C(a,b)=(a2+b2)/(a+b) are the harmonic, arithmetic, and contraharmonic means, respectively, and SHA(a,b)=SB(H,A), SAH(a,b)=SB(A,H), SCA(a,b)=SB(C,A), and SAC(a,b)=SB(A,C) are four Neuman means derived from the Schwab-Borchardt mean. |
| format | Article |
| id | doaj-art-fbf4ed29da7845d4b42c5512b0b85315 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
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| series | Journal of Applied Mathematics |
| spelling | doaj-art-fbf4ed29da7845d4b42c5512b0b853152025-02-03T01:07:29ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/807623807623Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic MeansZai-Yin He0Yu-Ming Chu1Miao-Kun Wang2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, ChinaSchool of Mathematics and Computation Science, Hunan City University, Yiyang 413000, ChinaSchool of Mathematics and Computation Science, Hunan City University, Yiyang 413000, ChinaFor a,b>0 with a≠b, the Schwab-Borchardt mean SB(a,b) is defined as SB(a,b)={b2-a2/cos-1(a/b) if a<b,a2-b2/cosh-1(a/b) if a>b. In this paper, we find the greatest values of α1 and α2 and the least values of β1 and β2 in [0,1/2] such that H(α1a+(1-α1)b,α1b+(1-α1)a)<SAH(a,b)<H(β1a+(1-β1)b,β1b+(1-β1)a) and H(α2a+(1-α2)b,α2b+(1-α2)a)<SHA(a,b)<H(β2a+(1-β2)b,β2b+(1-β2)a). Similarly, we also find the greatest values of α3 and α4 and the least values of β3 and β4 in [1/2,1] such that C(α3a+(1-α3)b,α3b+(1-α3)a)<SCA(a,b)<C(β3a+(1-β3)b,β3b+(1-β3)a) and C(α4a+(1-α4)b,α4b+(1-α4)a)<SAC(a,b)<C(β4a+(1-β4)b,β4b+(1-β4)a). Here, H(a,b)=2ab/(a+b), A(a,b)=(a+b)/2, and C(a,b)=(a2+b2)/(a+b) are the harmonic, arithmetic, and contraharmonic means, respectively, and SHA(a,b)=SB(H,A), SAH(a,b)=SB(A,H), SCA(a,b)=SB(C,A), and SAC(a,b)=SB(A,C) are four Neuman means derived from the Schwab-Borchardt mean.http://dx.doi.org/10.1155/2013/807623 |
| spellingShingle | Zai-Yin He Yu-Ming Chu Miao-Kun Wang Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means Journal of Applied Mathematics |
| title | Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means |
| title_full | Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means |
| title_fullStr | Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means |
| title_full_unstemmed | Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means |
| title_short | Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means |
| title_sort | optimal bounds for neuman means in terms of harmonic and contraharmonic means |
| url | http://dx.doi.org/10.1155/2013/807623 |
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