Composite entire functions with no unbounded Fatou components (Q2642158)

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 \textit{I. N. Baker} [J. Aust. Math. Soc., Ser. A 30, 483--495 (1981; Zbl 0474.30023)], \textit{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.