Proceedings - Mathematical Sciences

, Volume 122, Issue 1, pp 41–51

Optimal combinations bounds of root-square and arithmetic means for Toader mean

Article

Abstract

We find the greatest value α1 and α2, and the least values β1 and β2, such that the double inequalities α1S(a,b) + (1 − α1) A(a,b) < T(a,b) < β1S(a,b) + (1 − β1) A(a,b) and \(S^{\alpha_{2}}(a,b)A^{1-\alpha_{2}}(a,b)< T(a,b)< S^{\beta_{2}}(a,b)A^{1-\beta_{2}}(a,b)\) hold for all a,b > 0 with a ≠ b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, S(a,b) = [(a2 + b2)/2]1/2, A(a,b) = (a + b)/2, and \(T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta\) denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.

Keywords

Root-square mean arithmetic mean Toader mean complete elliptic integrals 

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References

  1. [1]
    Alzer H and Qiu S-L, Inequalities for means in two variables, Arch. Math. (Basel) 80(2) (2003) 201–215MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Alzer H, A power mean inequality for the gamma function, Monatsh. Math. 131(3) (2000) 179–188MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Alzer H and Qiu S-L, Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math. 172(2) (2004) 289–312MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Anderson G D, Vamanamurthy M K and Vuorinen M, Conformal Invariants, Inequalities, and Quasiconformal Maps (New York: John Wiley & Sons) (1997)MATHGoogle Scholar
  5. [5]
    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) 693–699MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Barnard R W, Pearce K and Richards K C, A monotonicity property involving 3 F 2 and comparisons of the classical approximations of elliptical arc length, SIAM J. Math. Anal. 32(2) (2000) 403–419MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Barnard R W, Pearce K and Schovanec L, Inequalities for the perimeter of an ellipse, J. Math. Anal. Appl. 260(2) (2001) 295–306MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Barnard R W, Richards K C and Tiedeman H C, A survey of some bounds for Gauss’s hypergeometric function and related bivariate means, J. Math. Inequal. 4(1) (2010) 45–52MathSciNetGoogle Scholar
  9. [9]
    Borwein J M and Borwein P B, Inequalities for compound mean iterations with logarithmic asymptotes, J. Math. Anal. Appl. 177(2) (1993) 572–582MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Bullen P S, Mitrinović D S and Vasić P M, Means and their inequalities (Dordrecht: D. Reidel Publishing Co.) (1988)MATHGoogle Scholar
  11. [11]
    Guo B-N and Qi F, Some bounds for the complete elliptic integrals of the first and second kinds, Math. Inequal. Appl. 14(2) (2011) 323–334MathSciNetMATHGoogle Scholar
  12. [12]
    Hardy G H, Littlewood J E and Pólya J E, Inequalities (Cambridge: Cambridge University Press) (1988)MATHGoogle Scholar
  13. [13]
    Hästö P A, Optimal inequalities between Seiffert’s mean and power means, Math. Inequal. Appl. 7(1) (2004) 47–53MathSciNetMATHGoogle Scholar
  14. [14]
    Huntington E V, Sets of independent postulates for the arithmetic mean, the geometric mean, the harmonic mean, and the root-mean-square, Trans. Am. Math. Soc. 29(1) (1927) 1–22MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Lin T P, The power mean and the logarithmic mean, Am. Math. Mon. 81 (1974) 879–883MATHCrossRefGoogle Scholar
  16. [16]
    Qiu S-L and Shen J-M, On two problems concerning means, J. Hangzhou Inst. Electronic Engg. 17(3) (1997) 1–7 (in Chinese)Google Scholar
  17. [17]
    Richards K C, Sharp power mean bounds for the Gaussian hypergeometric function, J. Math. Anal. Appl. 308(1) (2005) 303–313MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Toader Gh., Some mean values related to the arithmetic-geometric mean, J. Math. Anal. Appl. 218(2) (1998) 358–368MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Vamanamurthy M K and Vuorinen M, Inequalities for means, J. Math. Anal. Appl. 183(1) (1994) 155–166MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Vuorinen M, Hypergeometric functions in geometric function theory, Special functions and differential equations (Madras, 1997) (New Delhi: Allied Publ.) (1998) pp. 119–126Google Scholar
  21. [21]
    Yang Z-H, A new proof of inequalities for Gauss compound mean, Int. J. Math. Anal. 4(21) (2010) 1013–1018MathSciNetMATHGoogle Scholar
  22. [22]
    Zhang X-H, Wang G-D and Chu Y-M, Convexity with respect to Hölder mean involving zero-balanced hypergeometric functions, J. Math. Anal. Appl. 353(1) (2009) 256–259MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Computing ScienceHunan City UniversityYiyangChina
  2. 2.Department of MathematicsHuzhou Teachers CollegeHuzhouChina
  3. 3.Department of MathematicsZhejiang Sci-Tech UniversityHangzhouChina

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