Journal of the Mathematical Society of Japan

Julia sets of two permutable entire functions

Liangwen LIAO and Chung-Chun YANG

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In this paper first we prove that if f and g are two permutable transcendental entire functions satisfying f=f1(h) and g=g1(h), for some transcendental entire function h, rational function f1 and a function g1, which is analytic in the range of h, then F(g)F(f). Then as an application of this result, we show that if f(z)=p(z)eq(z)+c, where c is a constant, p a nonzero polynomial and q a nonconstant polynomial, or f(z)=zp(z)eq(z)dz, where p,q are nonconstant polynomials, such that f(g)=g(f) for a nonconstant entire function g, then J(f)=J(g).

Article information

J. Math. Soc. Japan, Volume 56, Number 1 (2004), 169-176.

First available in Project Euclid: 3 October 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58F23 30D35: Distribution of values, Nevanlinna theory

Julia set Fatou set permutable entire functions prime pseudo-prime


LIAO, Liangwen; YANG, Chung-Chun. Julia sets of two permutable entire functions. J. Math. Soc. Japan 56 (2004), no. 1, 169--176. doi:10.2969/jmsj/1191418700.

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