Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Further results on prime entire functions


Authors: Fred Gross and Chung Chun Yang
Journal: Trans. Amer. Math. Soc. 192 (1974), 347-355
MSC: Primary 30A20
DOI: https://doi.org/10.1090/S0002-9947-1974-0349972-3
MathSciNet review: 0349972
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let H denote the set of all the entire functions $ f(z)$ of the form: $ f(z) \equiv h(z){e^{p(z)}} + k(z)$ where $ p(z)$ is a nonconstant polynomial of degree m, and $ h(\nequiv\;0)$, $ k(\nequiv$ constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based on Nevanlinna's theory of meromorphic functions and some elementary facts about algebraic functions.


References [Enhancements On Off] (What's this?)

  • [1] 1. N. Baker and F. Gross, Further results on factorization of entire functions, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla. Calif., 1966), Amer. Math. Soc., Providence, R. I., 1968, pp. 30-35. MR 38 #6066. MR 0237785 (38:6066)
  • [2] R. P. Boas, Entire functions, Academic Press, New York, 1954. MR 16, 914. MR 0068627 (16:914f)
  • [3] P. Fatou, Sur l'itération des fonctions transcendantes entières, Acta Math. 47 (1926).
  • [4] R. Goldstein, On factorisation of certain entire functions, J. London Math. Soc. (2) 2 (1970), 221-224. MR 41 #2012. MR 0257361 (41:2012)
  • [5] -, On factorisation of certain entire functions. II, Proc. London Math. Soc. (3) 22 (1971), 483-506. MR 45 #546. MR 0291455 (45:546)
  • [6] -, Some results on factorization of meromorphic functions, J. London Math. Soc. (2) 4 (1971), 357-364. MR 44 #6963. MR 0289776 (44:6963)
  • [7] F. Gross, Prime entire functions, Trans. Amer. Math. Soc. 161 (1971), 219-233. MR 45 #547. MR 0291456 (45:547)
  • [8] -, On periodic left factors of meromorphic functions, Math. Research Center, Report 69-1, NRL Report 6963, U.S. Naval Research Laboratory, Washington, DC., 1969, 8 pp. MR 43 #6437. MR 0280718 (43:6437)
  • [9] -, Factorization of entire functions which are periodic $ \bmod\;g$, Indian J. Pure Appl. Math. 2 (1971).
  • [10] F. Gross and C. C. Yang, The fix-points and factorization of meromorphic functions, Trans. Amer. Math. Soc. 168 (1972), 211-219. MR 0301175 (46:333)
  • [11] W. K. Hayman, Meromorphic functions, Oxford Math. Monographs, Clarendon Press, Oxford, 1964. MR 29 #1337. MR 0164038 (29:1337)
  • [12] E. Hille, Analytic function theory. Vol. I, Introduction to Higher Math., Ginn, Boston, Mass., 1959, p. 267. MR 21 #6415. MR 0201608 (34:1490)
  • [13] M. Ozawa, On prime entire functions. 1 Kōdai Math. Sem. Rep. 22 (1970), 301-308; II. Kōdai Math. Sem. Rep 22 (1970), 309-312. MR 42 #4732. MR 0269837 (42:4732)
  • [14] A. Rényi and C. Rényi, Some remarks on periodic entire functions, J. Analyse Math. 14 (1965), 303-310. MR 31 #2406. MR 0178148 (31:2406)
  • [15] J. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922), 51-66. MR 1501189
  • [16] P. C. Rosenbloom, The fix-points of entire functions, Medd. Lunds Univ. Math. Sem., Tome Suppl., (1952), 186-192. MR 14, 546. MR 0051916 (14:546h)
  • [17] C. C. Yang, On the dependence of the zeros of an entire function and its factorization, J. Math. Anal. Appl. 35 (1971), 374-381. MR 0277715 (43:3448)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30A20

Retrieve articles in all journals with MSC: 30A20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0349972-3
Keywords: Factorization, factor, entire function, meromorphic function, prime function, pseudo-prime function, R-polynomial prime, Nevanlinna theory, E-prime
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society