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Composite entire functions with no unbounded Fatou components. (English) Zbl 1132.30017
Let $$f$$ be a transcendental entire function. It is known that the Julia set $$J(f)$$ is unbounded, so the Fatou set $$F(f)$$ cannot contain a neighborhood of $$\infty$$. In response to a question of I. N. Baker [J. Aust. Math. Soc., Ser. A 30, 483–495 (1981; Zbl 0474.30023)], Y. Wang [Isr. J. Math. 121, 55–60 (2001; Zbl 1054.37028)] showed that if $$f$$ has order less than $$1/2$$ but positive lower order, every component of $$F(f)$$ is bounded.
The paper under review considers a class of functions $$\Lambda$$ for which for $$\varepsilon> 0$$, $$\log L(r, f)> (1-\varepsilon)\log M(r, f)$$ for all $$r$$ outside a set of logarithmic density zero, and sets $$F= \bigcup_{k\geq 1} F_k$$ where $$F_k$$ is the set of transcendental entire functions for which $$\log\log M(r, f)\geq(\log r)^{1/k}$$. If $$h= f_{N^0} f_{N-1}\circ\cdots\circ f$$, where $$f_i$$ are in $$F\cap\Lambda$$ for $$i= 1,\dots N$$, then it is shown that $$h$$ has no unbounded Fatou component.

MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Fabry gaps
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References:
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