## 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:

 2.6e+61 Means
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