Proceedings - Mathematical Sciences

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

A double inequality for bounding Toader mean by the centroidal mean



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.


Toader mean centroidal mean complete elliptic integral double inequality 

2010 Mathematics Subject Classification

Primary: 26E60 Secondary: 26D20, 33E05 



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.


  1. [1]
    Alzer H and Qiu S-L, Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math. 172(2) (2004) 289–312; available online at
  2. [2]
    Anderson G D, Vamanamurthy M K and Vuorinen M, Conformal invariants, inequalities, and quasiconformal maps (1997) (New York: John Wiley & Sons)Google Scholar
  3. [3]
    Barnard R W, Pearce K and Richards K C, An inequality involving the generalized hypergeometric function and the arc length of an ellipse, SIAM J. Math. Anal. 31(3) (2000) 693–699; available online at
  4. [4]
    Bowman F, Introduction to elliptic functions with applications (1961) (New York: Dover Publications)Google Scholar
  5. [5]
    Byrd P F and Friedman M D, Handbook of elliptic integrals for engineers and scientists (1971) (New York: Springer-Verlag)Google Scholar
  6. [6]
    Chu Y-M and Wang M-K, Inequalities between arithmetic-geometric, Gini, and Toader means, Abstr. Appl. Anal. 2012, Article ID 830585, 11 pages; available online at
  7. [7]
    Chu Y-M and Wang M-K, Optimal Lehmer mean bounds for the Toader mean, Results Math. 61(3–4) (2012) 223–229; available online at
  8. [8]
    Chu Y-M, Wang M-K and Ma X-Y, Sharp bounds for Toader mean in terms of contraharmonic mean with applications, J. Math. Inequal. 7(2) (2013) 161–166; available online at
  9. [9]
    Chu Y-M, Wang M-K and Qiu S-L, Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci. 122(1) (2012) 41–51; available online at
  10. [10]
    Hua Y and Qi F, The best bounds for Toader mean in terms of the centroidal and arithmetic means, Filomat 28 (2014), in press; available online at
  11. [11]
    Hua Y and Qi F, The best bounds for Toader mean in terms of the centroidal and arithmetic means, available online at
  12. [12]
    Qiu S-L and Shen J-M, On two problems concerning means, J. Hangzhou Inst. Electron. Eng. 17(3) (1997) 1–7. (Chinese)Google Scholar
  13. [13]
    Toader Gh, Some mean values related to the arithmetic-geometric mean, J. Math. Anal. Appl. 218(2) (1998) 358–368;
  14. [14]
    Vuorinen M, Hypergeometric functions in geometric function theory, in: Special functions and differential equations, Proceedings of a Workshop held at The Institute of Mathematical Sciences, Madras, India, January 13–24, 1997 (1998) (New Delhi: Allied Publ.) pp. 119–126Google Scholar
  15. [15]
    Wang M-K, Chu Y-M, Qiu S-L and Jiang Y-P, Bounds for the perimeter of an ellipse, J. Approx. Theory 164(7) (2012) 928–937; available online at

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

Personalised recommendations