Some bounds for the complete elliptic integrals of the first and second kinds

In the article, the complete elliptic integrals of the first and second kinds are bounded by using the power series expansions of some functions, the celebrated Wallis' inequality, and an integral inequality due to R. P. Agarwal, P. Cerone, S. S. Dragomir and F. Qi.

[1]  Feng Qi (祁锋),et al.  The best bounds in Wallis' inequality , 2004 .

[2]  Zhang Guo-ming.  The Upper Bounds and Lower Bounds on Wallis′ Inequality , 2007 .

[3]  Qingbiao Wu,et al.  WALLIS INEQUALITY WITH A PARAMETER , 2006 .

[4]  Chao Chen.  On Wallis' inequality , 2006 .

[5]  Bruce C. Berndt,et al.  Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, p, and the Ladies Diary , 1988 .

[6]  Feng Qi (祁锋),et al.  Improvement of Lower Bound in Wallis’ Inequality , 2002 .

[7]  Feng Qi (祁锋).  Bounds for the Ratio of Two Gamma Functions , 2009 .

[8]  Feng Qi (祁锋),et al.  COMPLETELY MONOTONIC FUNCTION ASSOCIATED WITH THE GAMMA FUNCTIONS AND PROOF OF WALLIS' INEQUALITY , 2005 .

[9]  Á. Baricz.  Turán type inequalities for hypergeometric functions , 2008 .

[10]  Feng Qi (祁锋),et al.  A New Proof of the Best Bounds in Wallis' Inequality , 2003 .

[11]  Feng Qi (祁锋).  Inequalities for a Weighted Multiple Integral , 2001 .

[12]  Wu Qing-biao,et al.  Improvement on Wallis inequality , 2006 .

[13]  Feng Qi (祁锋).  BOUNDS FOR THE RATIO OF TWO GAMMA FUNCTIONS-FROM WENDEL'S AND RELATED INEQUALITIES TO LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS , 2009, 0904.1048.

[14]  Feng Qi (祁锋).  Inequalities for a Weighted Integral , 1999 .

[15]  Stamatis Koumandos,et al.  Remarks on a paper by Chao-Ping Chen and Feng Qi , 2005 .

[16]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[17]  S. Dragomir,et al.  LOBATTO TYPE QUADRATURE RULES FOR FUNCTIONS WITH BOUNDED DERIVATIVE , 2000 .

[18]  J. Gurland,et al.  On Wallis' Formula , 1956 .

[19]  Feng Qi (祁锋).  Inequalities for a Multiple Integral , 1999 .

[20]  Some New Iyengar Type Inequalities , 2005 .

[21]  Feng Qi (祁锋),et al.  Estimates for an integral in Lp norm of the (n+1)-th derivative of its integrand , 2000 .

[22]  Árpád Baricz,et al.  Turán type inequalities for generalized complete elliptic integrals , 2007 .

[23]  Feng Qi (祁锋),et al.  Some estimates of an integral in terms of the Lp-norm of the (n+1)st derivative of its integrand , 2003 .

[24]  G. N. Watson.  A Note on Gamma Functions , 1959 .

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

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

[27]  Zhao De-jun.  On a Two-sided Inequality Involving Wallis′s Formula , 2004 .

[28]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

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

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

[31]  ESTIMATE FOR UPPER BOUND OF AN ELLIPTIC INTEGRAL , 1996 .