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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Further results on fixpoints and zeros of entire functions

Authors: Jian Hua Zheng and Chung-Chun Yang
Journal: Trans. Amer. Math. Soc. 347 (1995), 37-50
MSC: Primary 30D05; Secondary 30D20
MathSciNet review: 1179403
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Abstract: In this paper, a quantitative estimation on the number of zeros of the function $ f \circ g(z) - \alpha (z)$ is derived, where $ f$ and $ g$ are transcendental entire functions and $ \alpha (z)$ a nonconstant polynomial. As an application of this and a further step towards an affirmative answer to a conjecture of Baker, a quantitative estimation on the number of period points of exact order $ n$ of $ {f_n}$ ($ n$th iterate of $ f$) is obtained.

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  • [1] Irvine Noel Baker, Zusammensetzungen ganzer Funktionen, Math. Z. 69 (1958), 121–163 (German). MR 0097532
  • [2] I. N. Baker, Fixpoints and iterates of entire functions, Math. Z. 71 (1959), 146–153. MR 0107015
  • [3] Irvine Noel Baker, The existence of fixpoints of entire functions, Math. Z. 73 (1960), 280–284. MR 0114008
  • [4] Walter Bergweiler, Proof of a conjecture of Gross concerning fix-points, Math. Z. 204 (1990), no. 3, 381–390. MR 1107470, 10.1007/BF02570881
  • [5] Walter Bergweiler, On the fix-points of composite functions, Pacific J. Math. 143 (1990), no. 1, 1–8. MR 1047396
  • [6] -, Periodic points of entire functions: Proof of a conjecture of Baker (to appear).
  • [7] J. Clunie, The composition of entire and meromorphic functions, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 75–92. MR 0271352
  • [8] F. Gross and C. F. Osgood, On fixed points of composite entire functions, J. London Math. Soc. (2) 28 (1983), no. 1, 57–61. MR 703464, 10.1112/jlms/s2-28.1.57
  • [9] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
  • [10] W. K. Hayman, On the characteristic of functions meromorphic in the plane and of their integrals, Proc. London Math. Soc. (3) 14a (1965), 93–128. MR 0180679
  • [11] W. K. Hayman, Research problems in function theory, The Athlone Press University of London, London, 1967. MR 0217268
  • [12] W. K. Hayman, Value distribution and A.P. gaps, J. London Math. Soc. (2) 28 (1983), no. 2, 327–338. MR 713387, 10.1112/jlms/s2-28.2.327
  • [13] R. Nevanlinna, Le théorème de Picard-Borel et la théorème des fonctions meromorphes, Gauthier-Villars, Paris, 1929.
  • [14] G. S. Prokopovich, Fixpoints of meromorphic functions, Ukrain. Mat. Zh. 25 (1973), 248-260; English transl., Ukrainian Math. J. pp. 198-208.
  • [15] P. C. Rosenbloom, The fixpoints of entire functions, Medd. Lunds Univ. Mat. Sem. [Tome Suppl.] (1952), 187-192.
  • [16] G. Valiron, Lectures on the general theory of integral functions, Edonard Privat, Toulouse, 1923.
  • [17] Le Yang, Zhi fenbu lun ji qi xin yanjiu, Chuncui Shuxue yu Yingyong Shuxue Zhuanzhu [Series of Monographs in Pure and Applied Mathematics], vol. 9, Kexue Chubanshe (Science Press), Beijing, 1982 (Chinese). MR 724784

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Article copyright: © Copyright 1995 American Mathematical Society