Regularity and fast escaping points of entire functions (Q2929546)

Let \(f\) be a transcendental entire function \(f\), and denote by \(f^n\), \(n=0,1,2, \dots\), the \(n\)th iterate of \(f\). The escaping set \(I(f)\) of \(f\) is defined as follows NEWLINE\[NEWLINE I(f)=\big\{z \;\big|\; f^n (z)\to \infty \text{ as } n\to \infty\big\}.NEWLINE\]NEWLINE \textit{A. E. Eremenko} [Banach Cent. Publ. 23, 339--345 (1989; Zbl 0692.30021)] proved that all components of \(\overline{I(f)}\) are unbounded and conjectured that all components of \({I(f)}\) itself are unbounded as well. Using the fast escaping set \(A(f)\) introduced by \textit{W. Bergweiler} and \textit{A. Hinkkanen} in [Math. Proc. Camb. Philos. Soc. 126, No. 3, 565--574 (1999; Zbl 0939.30019)], the authors proved in [Proc. Am. Math. Soc. 133, No. 4, 1119--1126 (2005; Zbl 1058.37033)] that \(I(f)\) has at least one unbounded component.NEWLINENEWLINELet \(M(r)=M(r,f)=\max\{ |f(z)|: |z|=r\}\), \(r>0\), and set NEWLINE\[NEWLINE A(f)=\big\{z : \exists l \in \mathbb{N}\; \big |\;f^{n+l}(z)|\geq M^n(R), \;n\in{\mathbb N}\big\},NEWLINE\]NEWLINE where \(M^n(r)\) denotes the iterations of \(M(r)\), and \(R>0\) is any value such that \(M(r)>r\) for \(r\geq R\). Let \(\mu_\varepsilon(r)=\varepsilon M(r)\), \(r>0\), \(0<\varepsilon<1\). Consider the sets NEWLINENEWLINENEWLINE\[NEWLINE Q_\varepsilon(f)=\big\{z : \exists l \in \mathbb{N}\;\big |\;f^{n+l}(z)|\geq \mu_\varepsilon^n(R),\; n\in\mathbb{N}\big\},NEWLINE\]NEWLINE where \(R>0\) is any value such that \(\mu_\varepsilon(r)>r\) for \(r\geq R\). We define the set \(Q(f)=\bigcup_{0<\varepsilon<1} Q_\varepsilon(f)\) to be the quite fast escaping set of \(f\). One has NEWLINE\[NEWLINE A(f)\subset Q_\varepsilon(f)\subset Q(f) \subset I(f), \quad 0<\varepsilon<1.NEWLINE\]NEWLINE The authors study the relations between \(A(f)\) and \(Q(f)\).NEWLINENEWLINE{Theorem 1.1.} Let \(f\) be a transcendental entire function in the Eremenko-Lubich class \(\mathcal{B}\). Then \(Q(f)=A(f)\).NEWLINENEWLINERecall that the class \(\mathcal{B}\) consists of transcendental entire functions whose set of singular values is bounded.NEWLINENEWLINELet \(R>0\) be any value such that \(M(r)>r\) for \(r\geq R\). The function \(f\) is said to be NEWLINENEWLINENEWLINENEWLINE\quad 1) \(\varepsilon\)-regular, where \(0<\varepsilon<1\), if \(\exists r=r(R)>0\) such that \(\mu_\varepsilon^n(r)\geq M^n(R)\); NEWLINENEWLINENEWLINE\quad 2) weakly regular if \(f\) is \(\varepsilon\)-regular for all \(0<\varepsilon<1\). NEWLINENEWLINENEWLINENEWLINE Theorem 1.2. Let \(f\) be a transcendental entire function and \(0<\varepsilon<1\). Then NEWLINENEWLINENEWLINE (1) \(f\) is regular if and only if \(Q_\varepsilon(f)=A(f)\); NEWLINENEWLINENEWLINE (2) \(f\) is weakly regular if and only if \(Q(f)=A(f)\). NEWLINENEWLINENEWLINENEWLINE Let \(m(r)=m(r,f)=\min\{ |f(z)|: |z|=r\}\), \(r>0\).NEWLINENEWLINE{ Theorem 1.3.} Let \(f\) be a transcendental entire function such that, for some \(r(f)>1\), NEWLINE\[NEWLINE m(r)\leq M(r)^{1- \frac{K}{\log r}} , \quad r\geq r(f),NEWLINE\]NEWLINE where \(K=4\log 4\). Then \(f\) is weakly regular, and hence \(Q(f)=A(f)\).NEWLINENEWLINEThe sets \(Q(f)\) and \(Q_\varepsilon\) have similar properties with respect to \(I(f)\). NEWLINENEWLINENEWLINENEWLINE { Theorem 2.1.} Let \(f\) be a transcendental entire function, and \(J(f)\) denote its Julia set. Then NEWLINE\[NEWLINE Q(f)\neq \emptyset, \quad Q(f)\cap J(f)\neq \emptyset, \quad J(f)=\overline{Q(f)\cap J(f)},\quad J(f)=\partial Q(f), NEWLINE\]NEWLINE and \(\overline{Q(f)}\) has no bounded components. Similar properties hold for each set \(Q_\varepsilon(f)\), where \(0<\varepsilon<1\).NEWLINENEWLINE{ Theorem 3.2.} Let \(0<\varepsilon <1\). There exists a transcendental entire function \(f\) such that \(Q_\varepsilon(f)\neq A(f)\) and, hence, \(Q(f)\neq A(f)\).NEWLINENEWLINEIn Section 4 the authors consider another regularity condition.