# A double inequality for bounding Toader mean by the centroidal mean

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## Abstract

In this paper, the authors find the best numbers for all

*α*and*β*such that$$\overline{C}\left(\alpha a+(1-\alpha)b,\alpha b+(1-\alpha)a\right)<T(a,b) <\overline{C}\left(\beta a+(1-\beta)b,\beta b+(1-\beta)a\right) $$

*a*,*b*>0 with*a*≠*b*, where \( \bar{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)}\) and \(T(a,b)=\frac {2}{\pi }{\int }_{0}^{{\pi }/{2}}\sqrt {a^{2}{\cos ^{2}{\theta }}+b^{2}{\sin ^{2}{\theta }}}\, \mathrm {d} \theta \) denote respectively the centroidal mean and Toader mean of two positive numbers*a*and*b*.## Keywords

Toader mean centroidal mean complete elliptic integral double inequality## 2010 Mathematics Subject Classification

Primary: 26E60 Secondary: 26D20, 33E05## Notes

### Acknowledgements

The authors would like to thank the anonymous referee for his/her careful reading and helpful corrections to the original version of this paper. The first author was partially supported by the Project of Shandong Province Higher Educational Science and Technology Program under Grant No. J11LA57, China.

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