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2013 Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means
Zai-Yin He, Yu-Ming Chu, Miao-Kun Wang
J. Appl. Math. 2013: 1-4 (2013). DOI: 10.1155/2013/807623


For a,b>0 with ab, 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.


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Zai-Yin He. Yu-Ming Chu. Miao-Kun Wang. "Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means." J. Appl. Math. 2013 1 - 4, 2013.


Published: 2013
First available in Project Euclid: 14 March 2014

zbMATH: 06950886
MathSciNet: MR3133966
Digital Object Identifier: 10.1155/2013/807623

Rights: Copyright © 2013 Hindawi


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