Landen inequalities for special functions

Baricz, Árpád

doi:10.1090/S0002-9939-2014-12016-8

Landen inequalities for special functions
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Proc. Amer. Math. Soc. 142 (2014), 3059-3066 Request permission

Abstract:

In this paper our aim is to present some Landen inequalities for Gaussian hypergeometric functions, confluent hypergeometric functions, generalized Bessel functions and general power series. Our main results complement and generalize some known results in the literature.

References

Gert Almkvist and Bruce Berndt, Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies diary, Amer. Math. Monthly 95 (1988), no. 7, 585–608. MR 966232, DOI 10.2307/2323302
Horst Alzer and Song-Liang Qiu, Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math. 172 (2004), no. 2, 289–312. MR 2095322, DOI 10.1016/j.cam.2004.02.009
G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl. 335 (2007), no. 2, 1294–1308. MR 2346906, DOI 10.1016/j.jmaa.2007.02.016
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
R. Balasubramanian, S. Ponnusamy, and M. Vuorinen, Functional inequalities for the quotients of hypergeometric functions, J. Math. Anal. Appl. 218 (1998), no. 1, 256–268. MR 1601889, DOI 10.1006/jmaa.1997.5776
Árpád Baricz, Landen-type inequality for Bessel functions, Comput. Methods Funct. Theory 5 (2005), no. 2, 373–379. MR 2205420, DOI 10.1007/BF03321104
Árpád Baricz, Bounds for modified Bessel functions of the first and second kinds, Proc. Edinb. Math. Soc. (2) 53 (2010), no. 3, 575–599. MR 2720238, DOI 10.1017/S0013091508001016
Árpád Baricz, Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, vol. 1994, Springer-Verlag, Berlin, 2010. MR 2656410, DOI 10.1007/978-3-642-12230-9
Mieczysław Biernacki and Jan Krzyż, On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 9 (1955), 135–147 (1957) (English, with Russian and Polish summaries). MR 89903
Ville Heikkala, Mavina K. Vamanamurthy, and Matti Vuorinen, Generalized elliptic integrals, Comput. Methods Funct. Theory 9 (2009), no. 1, 75–109. MR 2478265, DOI 10.1007/BF03321716
Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), no. 2, 278–301. MR 1600537, DOI 10.1112/S0025579300012602
S.-L. Qiu and M. Vuorinen, Landen inequalities for hypergeometric functions, Nagoya Math. J. 154 (1999), 31–56. MR 1689171, DOI 10.1017/S0027763000025290
Slavko Simić and Matti Vuorinen, Landen inequalities for zero-balanced hypergeometric functions, Abstr. Appl. Anal. , posted on (2012), Art. ID 932061, 11. MR 2898059, DOI 10.1155/2012/932061