[1] Toader G. Some mean values related to the arithmetic-geometric mean. J Math Anal Appl, 1998, 218(2):358-368 [2] Wang M K, Zhang W, Chu Y M. Monotonicity, convexity and inequalities involving the generalized elliptic integrals. Acta Math Sci, 2019, 39B(5):1440-1450 [3] Wang M K, Chu Y M. Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Mathematica Scientia, 2017, 37B(3):607-622 [4] Neuman E. On the calculation of elliptic integrals of the second and third kinds. Zastos Mat, 1969/1970, 11:91-94 [5] Neuman E. Elliptic integrals of the second and third kinds. Zastos Mat, 1969/1970, 11:99-102 [6] Byrd P F, Friedman M D. Handbook of Elliptic Integrals for Engineers and Scientists. New York:SpringerVerlag, 1971 [7] Prasolov V, Solovyev Y. Elliptic Functions and Elliptic Integrals. Providence:American Mathematical Society, 1997 [8] Neuman E. Bounds for symmetric elliptic integrals. J Approx Theory, 2003, 122(2):249-259 [9] Kazi H, Neuman E. Inequalities and bounds for elliptic integrals Ⅱ//Special Functions and Orthogonal Polynomials. Contemp Math, 471:127-138. Providence:Amer Math Soc, 2008 [10] Chu Y M, Wang M K, Qiu S L, Jiang Y P. Bounds for complete elliptic integrals of the second kind with applications. Comput Math Appl, 2012, 63(7):1177-1184 [11] Yang Z H, Qian W M, Chu Y M, Zhang W. On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. J Math Anal Appl, 2018, 462(2):1714-1726 [12] Zhao T H, Wang M K, Zhang W, Chu Y M. Quadratic transformation inequalities for Gaussian hypergeometric function. J Inequal Appl, 2018, 2018:Article 251 [13] Yang Z H, Qian W M, Chu Y M. Monotonicity properties and bounds involving the complete elliptic integrals of the first kind. Math Inequal Appl, 2018, 21(4):1185-1199 [14] Wang M K, Chu Y M, Zhang W. Monotonicity and inequalities involving zero-balanced hypergeometric functions. Math Inequal Appl, 2019, 22(2):601-617 [15] Qiu S L, Ma X Y, Chu Y M. Sharp Landen transformation inequalities for hypergeometric functions, with applications. J Math Anal Appl, 2019, 474(2):1306-1337 [16] Yang Z H, Qian W M, Zhang W, Chu Y M. Notes on the complete elliptic integral of the first kind. Math Inequal Appl, 2020, 23(1):77-93 [17] Wang M K, He Z Y, Chu Y M. Sharp power mean inequalities for the generalized elliptic integral of the first kind. Comput Methods Funct Theory, 2020, 20(1):111-124 [18] Qian W M, He Z Y, Chu Y M. Approximation for the complete elliptic integral of the first kind. Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM, 2020, 114(2), Article 57. https://doi.org/10.1007/s13398-020-00784-9 [19] Anderson G D, Vamanamurthy M K, Vuorinen M. Conformal Invariants, Inequalities, and Quasiconformal Maps. New York:John Wiley & Sons, 1997 [20] Wang M K, Hong M Y, Xu Y F, et al. Inequalities for generalized trigonometric and hyperbolic functions with one parameter. J Math Inequal, 2020, 14(1):1-21 [21] Huang T R, Tan S Y, Ma X Y, Chu Y M. Monotonicity properties and bounds for the complete p-elliptic integrals. J Inequal Appl, 2018, 2018:Article 239 [22] Vuorinen M. Hypergeometric functions in geometric function theory//Special Functions and Differential Equations. Madras, 1997:119-126; New Delhi:Allied Publ, 1998 [23] Barnard R W, Pearce K, Richards K C. A monotonicity property involving 3F2 and comparisons of the classical approximations of elliptical arc length. SIAM J Math Anal, 2000, 32(2):403-419 [24] Alzer H, Qiu S L. Monotonicity theorems and inequalities for the complete elliptic integrals. J Comput Appl Math, 2004, 172(2):289-312 [25] Kazi H, Neuman E. Inequalities and bounds for elliptic integrals. J Approx Theory, 2007, 146(2):212-226 [26] Atkinson K E. An Introduction to Numerical Analysis. New York:John Wiley & Songs, 1989 [27] Chu Y M, Wang M K, Qiu S L, Qiu Y F. Sharp generalized Seiffert mean bounds for Toader mean. Abstr Appl Anal, 2011, 2011:Article ID 605259 [28] Chu Y M, Wang M K, Qiu S L. Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc Indian Acad Sci Math Sci, 2012, 122(1):41-51 [29] Chu Y M, Wang M K. Optimal Lehmer mean bounds for the Toader mean. Results Math, 2012, 61(3/4):223-229 [30] Wang M K, Chu Y M, Qiu S L, Jiang Y P. Bounds for the perimeter of an ellipse. J Approx Theory, 2012, 164(7):928-937 [31] Wang M K, Chu Y M. Asymptotical bounds for complete elliptic integrals of the second kind. J Math Anal Appl, 2013, 402(1):119-126 [32] Chu Y M, Wang M K, Ma X Y. Sharp bounds for Toader mean in terms of contraharmonic mean with applications. J Math Inequal Appl, 2013, 7(2):161-166 [33] Li J F, Qian W M, Chu Y M. Sharp bounds for Toader mean in terms of arithmetic, quadratic, and Neuman means. J Inequal Appl, 2015, 2015:Article 277 [34] Chu H H, Qian W M, Chu Y M, Song Y Q. Optimal bounds for a Toader-type mean in terms of oneparameter quadratic and contraharmonic means. J Nonlinear Sci Appl, 2016, 9(5):3424-3432 [35] Neuman E. Inequalities and bounds for generalized complete elliptic integrals. J Math Anal Appl, 2011, 373(1):203-213 [36] Chu Y M, Wang M K. Inequalities between arithmetic-geometric, Gini, and Toader means. Abstr Appl Anal, 2012, 2012:Article ID 830585 [37] Xia W F, Chu Y M. The Schur convexity of Gini mean values in the sense of harmonic mean. Acta Mathematica Scientia, 2011, 31B(3):1103-1112 [38] Neuman E. Inequalities for Jacobian elliptic functions and Gauss lemniscate functions. Appl Math Comput, 2012, 218(15):7774-7782 [39] Hua Y, Qi F. A double inequality for bounding Toader mean by the centroidal mean. Proc Indian Acad Sci Math Sci, 2014, 124(4):727-531 [40] Hua Y, Qi F. The best bounds for Toader mean in terms of the centroidal and arithmetic means. Filomat, 2014, 28(4):775-780 [41] Neuman E. Inequalities involving generalized Jacobian elliptic functions. Integral Transforms Spec Funct, 2014, 25(11):864-873 [42] Neuman E. Inequalities for the generalized trigonometric, hyperbolic and Jacobian elliptic functions. J Math Inequal, 2015, 9(3):709-726 [43] Jiang W D, Qi F. A double inequality for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean. Publ Inst Math, 2016, 99(113):237-242 [44] Zhao T H, Chu Y M, Zhang W. Optimal inequalities for bounding Toader mean by arithmetic and quadratic means. J Inequal Appl, 2017, 2017:Article 26 [45] Wang, J L, Qian W M, He Z Y, Chu Y M. On approximating the Toader mean by other bivariate means. J Funct Spaces, 2019, 2019:Article 6082413 |