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Optimal bounds for Neuman means in terms of harmonic and contraharmonic means. (English) Zbl 1397.26015

Summary: For \(a,v>0\) with \(a\neq b\), the Schwab-Borchardt mean \(\mathrm{SB}(a,b)\) is defined as \(\mathrm{SB}(a,b)=\{\sqrt{b^2-a^2}/\cos^{-1}(a/b)\) if \(a<b\), \(\sqrt{b^2-a^2}/\cosh^{-1}(a/b)\) if \(a>b\). In this paper, we find the greatest values of \(\alpha_1\) and \(\alpha_2\) and the least values of \(\beta_1\) and \(\beta_2\) in \([0,1/2]\) such that \(H(\alpha_1a+(1-\alpha_1)b,\alpha_1b+(1-\alpha_1)a)<\mathrm{S}_{AH}(a,b)<H(\beta_1a+(1-\beta_1)b,\beta_1b+(1-\beta_1)a)\) and \(H(\alpha_2a+(1-\alpha_2)b,\alpha_2b+(1-\alpha_2)a)<\mathrm{S}_{HA}(a,b)<H(\beta_2a+(1-\beta_2)b,\beta_2b+(1-\beta_2)a)\). Similarly, we also find the greatest values of \(\alpha_3\) and \(\alpha_4\) and the least values of \(\beta_3\) and \(\beta_4\) in \([1/2,1]\) such that \(C(\alpha_3a+(1-\alpha_3)b,\alpha_3b+(1-\alpha_3)a)<\mathrm{S}_{CA}(a,b)<C(\beta_3a+(1-\beta_3)b,\beta_3b+(1-\beta_3)a)\) and \(C(\alpha_4a+(1-\alpha_4)b,\alpha_4b+(1-\alpha_4)a)<\mathrm{S}_{AC}(a,b)<C(\beta_4a+(1-\beta_4)b,\beta_4b+(1-\beta_4)a)\). Here, \(H(a,b)=2ab/(a+b)\), \(A(a,b)=( a+b)/2\), and \(C(a,b)=(a^2+b^2)/(a+b)\) are the harmonic, arithmetic, and contraharmonic means, respectively, and \(\mathrm{S}(HA)(a,b)=\mathrm{SB}(H,A)\), \(\mathrm{S}(AH)(a,b)=\mathrm{SB}(A,H)\), \(\mathrm{S}(CA)(a,b)=\mathrm{SB}(C,A)\), and \(\mathrm{S}(AC)(a,b)=\mathrm{SB}(A,C)\) are four Neuman means derived from the Schwab-Borchardt mean.

MSC:

26E60 Means
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