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Proceedings - Mathematical Sciences

, Volume 124, Issue 4, pp 527–531 | Cite as

A double inequality for bounding Toader mean by the centroidal mean

  • YUN HUA
  • FENG QI
Article

Abstract

In this paper, the authors find the best numbers α and β such that
$$\overline{C}\left(\alpha a+(1-\alpha)b,\alpha b+(1-\alpha)a\right)<T(a,b) <\overline{C}\left(\beta a+(1-\beta)b,\beta b+(1-\beta)a\right) $$
for all a,b>0 with ab, 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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Copyright information

© Indian Academy of Sciences 2014

Authors and Affiliations

  1. 1.Department of Information EngineeringWeihai Vocational CollegeWeihai CityChina
  2. 2.College of MathematicsInner Mongolia University for NationalitiesTongliao CityChina

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