Một giới hạn dưới chính xác cho các tích phân elliptic toàn phần loại một

Borwein, J.M., Borwein, P.B.: Pi and the AGM. Wiley, New York (1987) Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, London (1958) Carlson, B.C., Vuorinen, M.: Inequality of the AGM and the logarithmic mean. SIAM. Rev. 33, 655 (1991). (Problem 91–117) Todd, J., Braden, B., Danloy, B., Schmidt, F.: Inequality of the AGM and the logarithmic Mean (B. C. Carlson and M. Vuorinen). SIAM. Rev. 34(4), 653–654 (1992) Sándor, J.: On certain inequalities for means. J. Math. Anal. Appl. 189, 602–606 (1995) Bracken, P.: An arithmetic–geometric mean inequality. Exposition. Math. 19, 273–279 (2001) Neuman, E., Sándor, J.: On certain means of two arguments and their extensions. Int. J. Math. Math. Sci. 16, 981–993 (2003) Qi, F., Sofo, A.: An alternative and united proof of a double inequality for bounding the arithmetic-geometric mean. http://arxiv.org/pdf/0902.2515v1.pdf Yang, Z.-H.: A new proof of inequalities for Gauss compound mean. Int. J. Math. Anal. 4(21), 1013–1018 (2010) Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23, 512–524 (1992) Vamanamurthy, M.K., Vuorinen, M.: Inequalities for means. J. Math. Anal. Appl. 183(1), 155–166 (1994) Borwein, J.M., Borwein, P.B.: Inequalities for compound mean iterations with logarithmic asymptotes. J. Math. Anal. Appl. 177(2), 572–582 (1993) Yang, Z.-H., Song, Y.-Q., Chu, Y.-M.: Sharp bounds for the arithmetic-geometric mean. J. Inequal. Appl. 2014, 192 (2014) Alzer, H., Qiu, S.-L.: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172, 289–312 (2004) 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. 462, 1714–1726 (2018) 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. 114(2), 12 (2020). (Article ID 57) Borwein, J.M., Borwein, P.B.: The arithmetic–geometric mean and fast computation of elementary functions. SIAM Rev. 26, 351–366 (1984) Carlson, B.C., Guatafson, J.L.: Asymptotic expansion of the first elliptic integral. SIAM J. Math. Anal. 16(5), 1072–1092 (1985) Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21, 536–549 (1990) Kühnau, R.: Eine Methode, die Positivitä t einer Funktion zu prüfen. Z. Angew. Math. Mech. 74, 140–143 (1994). (in German) Qiu, S.-L., Vamanamurthy, M.K.: Sharp estimates for complete elliptic integrals. SIAM. J. Math. Anal. 27(3), 823–834 (1996) Alzer, H.: Sharp inequalities for the complete elliptic integral of the first kind. Math. Proc. Camb. Philos. Soc. 124(2), 309–314 (1998) Qiu, S.-L., Vuorinen, M.: Landen inequalities for hypergeometric functions. Nagoya Math. J. 154, 31–56 (1999) Wang, M.-K., Chu, Y.-M., Qiu, S.-L.: Some monotonicity properties of generalized ellipitic integrals with applications. Math. Inequal. Appl. 16(3), 671–677 (2013) Wang, M.-K., Chu, Y.-M., Qiu, S.-L.: Sharp bounds for generalized elliptic integrals of the first kind. J. Math. Anal. Appl. 429, 744–757 (2015) Yang, Z.-H., Chu, Y.-M.: A monotonicity property involving the generalized elliptic integrals of the first kind. Math. Inequal. Appl. 20(3), 729–735 (2017) 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. 21(4), 1185–1199 (2018) Wang, M.K., Zhang, W., Chu, Y.M.: Monotonicity, convexity and inequalities involving the generalized elliptic integrals. Acta Math. Sci. Ser. B (Engl. Ed.) 39(5), 1440–1450 (2019) Yang, Z.-H., Tian, J.: Sharp inequalities for the generalized elliptic integrals of the first kind. Ramanujan J. 48, 91–116 (2019) Yang, Z.-H., Tian, J.: Convesity and monotonicity for elliptic integrals of the first kind and applications. Appl. Anal. Discret. Math. 13, 240–260 (2019) Wang, M.K., Chu, Y.M., Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 22(2), 601–617 (2019) Wang, M.-K., Zhang, W., Chu, Y.-M.: Monotonicity, convexity and inequalities involving the generalized elliptic integrals. Acta Math. Sci. 39B(5), 1440–1450 (2019) Wang, M.-K., Chu, H.-H., Li, Y.-M., Chu, Y.-M.: Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the firrst kind. Appl. Anal. Discret. Math. 14(1), 255–271 (2020) Yang, Z.-H., Qian, W.-M., Zhang, W., Chu, Y.-M.: Notes on the complete elliptic integral of the first kind. Math. Inequal. Appl. 23(1), 77–93 (2020) Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997) Ponnusamy, S., Vuorinen, M.: Asymptotic expansions and inequalities for hypergeometric functions. Mathematika 44, 278–301 (1997) Heikkala, V., Lindén, H., Vamanamurthy, M.K., Vuorinen, M.: Generalized elliptic integrals and the Legendre M-function. J. Math. Anal. Appl. 338(2), 223–243 (2008) Heikkala, V., Vamanamurthy, M.K., Vuorinen, M.: Generalized elliptic integrals. Comput. Methods Funct. Theory 9(1), 75–109 (2009) Neuman, E.: Inequalities and bounds for generalized complete elliptic integrals. J. Math. Anal. Appl. 373(1), 203–213 (2011) Wang, M.-K., Chu, Y.-M., Jiang, Y.-P.: Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mt. J. Math. 46(2), 679–691 (2016) 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 20(1), 111–124 (2020) Zhao, T.-H., Wang, M.-K., Chu, Y.-M.: A sharp double inequality involving generalized complete elliptic integral of the first kind. AIMS Math. 5(5), 4512–4528 (2020) Belzunce, F., Ortega, E., Ruiz, J.M.: On non-monotonic ageing properties from the Laplace transform, with actuarial applications. Insurance Math. Econom. 40, 1–14 (2007) Yang, Z.-H., Chu, Y.-M., Tao, X.-J.: A double inequality for the trigamma function and its applications. Abstr. Appl. Anal. 9, 2014 (2014). (Art, ID 702718 ) Yang, Z.-H., Chu, Y.-M., Zhang, X.-H.: Necessary and sufficient conditions for functions involving the psi function to be completely monotonic. J. Inequal. Appl. 2015, 157 (2015) Yang, Z.-H., Chu, Y.-M.: Inequalities for certain means in two arguments. J. Inequal. Appl. 2015, 299 (2015) Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, 210 (2017) Yang, Z.-H., Tian, J.: Monotonicity and sharp inequalities related to gamma function. J. Math. Inequal. 12(1), 1–22 (2018) Yang, Z.-H., Tian, J.-F.: Windschitl type approximation formulas for the gamma function. J. Inequal. Appl. 2018, 272 (2018) Yang, Z.-H., Tian, J.-F.: An accurate approximation formula for gamma function. J. Inequal. Appl. 2018, 56 (2018) Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the error function. Math. Inequal. App. 21(2), 469–479 (2019) Yang, Z.-H.: Sharp approximations for the complete elliptic integrals of the second kind by one-parameter means. J. Math. Anal. Appl. 467, 446–461 (2018) Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019) Lv, H.-L., Yang, Z.-H., Zheng, S.: Monotonicity and inequalities involving the incomplete gamma function. J. Math. Inequal. 13(2), 351–367 (2019) Yang, Z., Tian, J.-F.: Monotonicity rules for the ratio of two Laplace transforms with applications. J. Math. Anal. Appl. 470, 821–845 (2019) Yang, Z.-H., Tian, J.: Convesity and monotonicity for elliptic integrals of the first kind and applications. https://arxiv.org/abs/1705.05703 Yang, Z.-H., Tian, J.-F., Zhu, Y.-R.: A rational approximation for the complete elliptic integrals. Math. 8, 635 (2020)