## Journal of the Mathematical Society of Japan

### Julia sets of two permutable entire functions

#### Abstract

In this paper first we prove that if $f$ and $g$ are two permutable transcendental entire functions satisfying $f=f_1(h)$ and $g=g_{1}(h)$, for some transcendental entire function $h$, rational function $f_1$ and a function $g_{1}$, which is analytic in the range of $h$, then $F(g)\subset F(f)$. Then as an application of this result, we show that if $f(z)=p(z)e^{q(z)}+c$, where $c$ is a constant, $p$ a nonzero polynomial and $q$ a nonconstant polynomial, or $f(z)=\int^{z}p(z)e^{q(z)}dz$, 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

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

Dates
First available in Project Euclid: 3 October 2007

https://projecteuclid.org/euclid.jmsj/1191418700

Digital Object Identifier
doi:10.2969/jmsj/1191418700

Mathematical Reviews number (MathSciNet)
MR2027620

Zentralblatt MATH identifier
1049.37036

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

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1191418700