Further results on fixpoints and zeros of entire functions

Zheng, Jian Hua; Yang, Chung-Chun

doi:10.1090/S0002-9947-1995-1179403-9

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
DOI: https://doi.org/10.1090/S0002-9947-1995-1179403-9
MathSciNet review: 1179403
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

Irvine Noel Baker, Zusammensetzungen ganzer Funktionen, Math. Z. 69 (1958), 121–163 (German). MR 97532, DOI https://doi.org/10.1007/BF01187396
I. N. Baker, Fixpoints and iterates of entire functions, Math. Z. 71 (1959), 146–153. MR 107015, DOI https://doi.org/10.1007/BF01181396
I. N. Baker, Some entire functions with fixpoints of every order, J. Austral. Math. Soc. 1 (1959/1961), 203–209. MR 0114007
Walter Bergweiler, Proof of a conjecture of Gross concerning fix-points, Math. Z. 204 (1990), no. 3, 381–390. MR 1107470, DOI https://doi.org/10.1007/BF02570881
Walter Bergweiler, On the fix-points of composite functions, Pacific J. Math. 143 (1990), no. 1, 1–8. MR 1047396

---, Periodic points of entire functions: Proof of a conjecture of Baker (to appear).

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
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, DOI https://doi.org/10.1112/jlms/s2-28.1.57
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
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 180679, DOI https://doi.org/10.1112/plms/s3-14A.1.93
W. K. Hayman, Research problems in function theory, The Athlone Press University of London, London, 1967. MR 0217268
W. K. Hayman, Value distribution and A.P. gaps, J. London Math. Soc. (2) 28 (1983), no. 2, 327–338. MR 713387, DOI https://doi.org/10.1112/jlms/s2-28.2.327

R. Nevanlinna, Le théorème de Picard-Borel et la théorème des fonctions meromorphes, Gauthier-Villars, Paris, 1929.

G. S. Prokopovich, Fixpoints of meromorphic functions, Ukrain. Mat. Zh. 25 (1973), 248-260; English transl., Ukrainian Math. J. pp. 198-208.

P. C. Rosenbloom, The fixpoints of entire functions, Medd. Lunds Univ. Mat. Sem. [Tome Suppl.] (1952), 187-192.

G. Valiron, Lectures on the general theory of integral functions, Edonard Privat, Toulouse, 1923.

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