On the radial distribution of Julia sets of entire solutions of \(f^{(n)} + A(z)f = 0\)
For a transcendental entire (or meromorphic) function \(f\) denote by \(J(f)\) the Julia set of \(f\). The set \(\{ z\in\mathbb{C}: \arg z=\theta\}\), \(\theta\in[0,2\pi)\), is said to be a radial distribution of \(f\) if \(\Omega(\theta-\varepsilon,\theta+\varepsilon)\cap J(f)\) is unbounded for every \(\varepsilon>0\), where, for \(0\leq\alpha <\beta<2\pi\), the set \(\Omega(\alpha,\beta)\) denotes the open sector bounded by the rays \(\arg z=\alpha\) and \(\arg z=\beta\). Denote by \(\Delta(f)\) the set of all such angles for \(f\). The study of the linear measure \(\operatorname{mes} \Delta(f)\) of \(\Delta(f)\) has a long history. \textit{I. N. Baker} [J. Lond. Math. Soc. 40, 499--502 (1965; Zbl 0136.37804)] constructed an entire function with infinite lower order for which the Julia set is contained in a horizontal strip. On the other hand, it is proved in [\textit{J. Qiao}, Acta Math. Sin. 37, No. 5, 702--708 (1994; Zbl 0814.30019)] that for \(f\) transcendental entire with finite lower order \(\mu(f)\), \(\operatorname{mes}\Delta(f) = 2\pi\) if \(\mu(f)\) is less than \(1/2\) and \(\operatorname{mes}\Delta(f) \geq \pi/\mu(f)\) otherwise. Hence it is interesting to study \(\operatorname{mes}\Delta(f)\) for functions with infinite lower order. In the article under review, the authors restrict to the solutions of the differential equation \[ f^{(n)}(z) + A(z) f(z) = 0, \] where \(n\geq 2\) and \(A\) is transcendental entire. It is not hard to see that the nontrivial solutions of this equation are transcendental entire functions of infinite lower order. As the main result, it is proved that if \(E\) is a finite product of solutions to the above equation, then \[ \operatorname{mes} \Delta(E)\geq \min\left\{2\pi, \frac{\pi}{\sigma(A)}\right\}, \] where \(\sigma\) denotes the usual order of growth. If \(\operatorname{mes}\Delta(E)=2\pi\), this result implies that \(E\) has no unbounded Fatou components.
scientific article; zbMATH DE number 5994366
scientific article