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Landen inequalities for special functions


Author: Árpád Baricz
Journal: Proc. Amer. Math. Soc. 142 (2014), 3059-3066
MSC (2010): Primary 39B62, 33C05
DOI: https://doi.org/10.1090/S0002-9939-2014-12016-8
Published electronically: May 8, 2014
MathSciNet review: 3223362
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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.


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  • [1] 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 (89j:01028), https://doi.org/10.2307/2323302
  • [2] 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 (2005i:33021), https://doi.org/10.1016/j.cam.2004.02.009
  • [3] 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 (2008i:33012), https://doi.org/10.1016/j.jmaa.2007.02.016
  • [4] 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 (2000g:33001)
  • [5] 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 (99k:33007), https://doi.org/10.1006/jmaa.1997.5776
  • [6] Árpád Baricz, Landen-type inequality for Bessel functions, Comput. Methods Funct. Theory 5 (2005), no. 2, 373-379. MR 2205420 (2008i:33019), https://doi.org/10.1007/BF03321104
  • [7] 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 (2011k:33010), https://doi.org/10.1017/S0013091508001016
  • [8] Árpád Baricz, Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, vol. 1994, Springer-Verlag, Berlin, 2010. MR 2656410 (2011f:33007)
  • [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 Polish and Russian summaries). MR 0089903 (19,736f)
  • [10] Ville Heikkala, Mavina K. Vamanamurthy, and Matti Vuorinen, Generalized elliptic integrals, Comput. Methods Funct. Theory 9 (2009), no. 1, 75-109. MR 2478265 (2009k:33016), https://doi.org/10.1007/BF03321716
  • [11] NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, 2010. Edited by Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W. Clark; with 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248 (2012a:33001)
  • [12] S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), no. 2, 278-301. MR 1600537 (99b:33004), https://doi.org/10.1112/S0025579300012602
  • [13] S.-L. Qiu and M. Vuorinen, Landen inequalities for hypergeometric functions, Nagoya Math. J. 154 (1999), 31-56. MR 1689171 (2001g:33004)
  • [14] Slavko Simić and Matti Vuorinen, Landen inequalities for zero-balanced hypergeometric functions, Abstr. Appl. Anal. (2012), Art. ID 932061, 11. MR 2898059

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Additional Information

Árpád Baricz
Affiliation: Department of Economics, Babeş-Bolyai University, Cluj-Napoca 400591, Romania
Email: bariczocsi@yahoo.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12016-8
Keywords: Hypergeometric functions, power series, Landen inequalities.
Received by editor(s): August 23, 2012
Received by editor(s) in revised form: September 10, 2012
Published electronically: May 8, 2014
Communicated by: Walter Van Assche
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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