The best bounds for Toader mean in terms of the centroidal and arithmetic means

In the paper, the authors discover the best constants ?1, ?2, ?1, and ?2 for the double inequalities ?1C(a,b) + (1-?1)A(a,b) < T(a,b) < ?1C(a,b) + (1-?1)A(a,b) and ?2/A(a,b) + 1-?2/C(a,b) < 1/T(a,b) < ?2/A(a,b) + 1-?2-C(a,b) to be valid for all a, b > 0 with a ? b, where C(a,b) = 2(a2+ab+b2)/3(a+b), A(a,b) = a+b/2, and T(a,b) = 2/???2,0 ?a2 cos2 ? + b2 sin2 ? d ? are respectively the centroidal, arithmetic, and Toader means of two positive numbers a and b. As an application of the above inequalities, the authors also find some new bounds for the complete elliptic integral of the second kind.

[1]  C. Berg,et al.  Complete Monotonicity of a Difference Between the Exponential and Trigamma Functions and Properties Related to a Modified Bessel Function , 2013 .

[2]  Feng Qi (祁锋).  Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions , 2013, 1302.6731.

[3]  Feng Qi,et al.  Some inequalities for complete elliptic integrals , 2013, 1301.4385.

[4]  Yue-Ping Jiang,et al.  Concavity of the complete elliptic integrals of the second kind with respect to Hölder means , 2012 .

[5]  YU-MING CHU,et al.  Optimal combinations bounds of root-square and arithmetic means for Toader mean , 2012 .

[6]  Árpád Baricz,et al.  Bounds for modified Bessel functions of the first and second kinds , 2010, Proceedings of the Edinburgh Mathematical Society.

[7]  Henrik L. Pedersen,et al.  On the asymptotic expansion of the logarithm of Barnes triple gamma function , 2009 .

[8]  Feng Qi (祁锋),et al.  Refinements, Generalizations, and Applications of Jordan's Inequality and Related Problems , 2009 .

[9]  Feng Qi,et al.  Some bounds for the complete elliptic integrals of the first and second kinds , 2009, 0905.2787.

[10]  Feng Qi (祁锋),et al.  An alternative and united proof of a double inequality bounding the arithmetic-geometric mean , 2009, 0902.2515.

[11]  T. N. Shanmugam,et al.  Hypergeometric functions in the geometric function theory , 2007, Appl. Math. Comput..

[12]  Iosif Pinelis,et al.  L'Hospital Rules for Monotonicity and the Wilker-Anglesio Inequality , 2004, Am. Math. Mon..

[13]  Horst Alzer,et al.  Monotonicity theorems and inequalities for the complete elliptic integrals , 2004 .

[14]  E. Weisstein.  Complete Elliptic Integral of the First Kind , 2002 .

[15]  Roger W. Barnard,et al.  An Inequality Involving the Generalized Hypergeometric Function and the Arc Length of an Ellipse , 2000, SIAM J. Math. Anal..

[16]  Gh. Toader,et al.  Some Mean Values Related to the Arithmetic–Geometric Mean , 1998 .

[17]  Matti Vuorinen,et al.  Asymptotic expansions and in-equalities for hypergeometric functions , 1997 .

[18]  G. Anderson,et al.  Conformal Invariants, Inequalities, and Quasiconformal Maps , 1997 .

[19]  Matti Vuorinen,et al.  Functional inequalities for hypergeometric functions and complete elliptic integrals , 1992 .

[20]  Q. I. Rahman,et al.  On the Monotonicity of Certain Functionals in the Theory of Analytic Functions , 1967, Canadian Mathematical Bulletin.

[21]  J. M. Thomas,et al.  Conformal Invariants. , 1926, Proceedings of the National Academy of Sciences of the United States of America.

[22]  褚玉明,et al.  Convexity of the complete elliptic ingegrals of the first kind with respect to Holder means , 2012 .

[23]  Á. Baricz,et al.  Bounds for complete elliptic integrals of the first kind , 2010 .

[24]  Tomohiro Hayashi.  ARITHMETIC-GEOMETRIC MEAN INEQUALITY , 2009 .

[25]  Matti Vuorinen,et al.  Generalized elliptic integrals and modular equations , 2000 .

[26]  Feng Qi (祁锋),et al.  Some inequalities constructed by Tchebysheff's integral inequality , 1999 .

[27]  Paul F. Byrd,et al.  Handbook of elliptic integrals for engineers and scientists , 1971 .

[28]  F. Bowman.  Introduction to Elliptic Functions with Applications , 1961 .