How to detect Hayman directions (Q2378567)

In order to answer the question in the title of this paper, the author introduces a class of \textit{Hayman exceptional functions}. He expects that it plays a similar role for the study of Hayman directions as Julia exceptional functions do for Julia directions, a characterization of meromorphic functions without any Hayman directions by means of this class. A \textit{Hayman direction} of a meromorphic function \(f:\mathbb{C} \to \hat{\mathbb{C}}\) is a ray from the origin to the point at infinity in the plane \(\mathbb{C}\), around which Hayman's theorem holds for \(f\), that is, for every positive integer \(k\in\mathbb{N}\), either \(f\) assumes every finite value infinitely often or \(f^{(k)}\) assumes every finite value except zero infinitely often. \textit{Lo Yang} [J. Lond. Math. Soc., II. Ser. 25, 288--296 (1982; Zbl 0458.30019)] has introduced this notion and confirmed its existence under the assumption \((T3)\) \[ \limsup_{r\to\infty} \frac{ T(r,f) }{ (\log r)^3 }=\infty. \] Two results motivated him: one is of course Hayman's theorem for transcendental meromorphic functions in~\(\mathbb{C}\), found by \textit{W. K. Hayman} himself in [Ann. Math. (2) 70, 9--42 (1959; Zbl 0088.28505)], and a criterion for normality of the family of meromorphic functions defined on a domain in \(\mathbb{C}\), proved by \textit{Y.-X. Ku} [Sci. Sinica, Special Issue I on Math. 267--274 (1979; Zbl 1171.30308)]. The most familiar such marriage is performed by Julia directions, Picard's theorem and Montel's criterion. Then the matchmaker is the notion of ``cercles de remplissage'' or ``filling disks'' of a transcendental meromorphic function \(f\), which is a sequence of disks of the form \(C_j=\{z: |z-z_j|<\varepsilon_j |z_j|\}\) with \(z_j\to\infty\), \(\varepsilon_j\to 0\) as \(j\to\infty\). The requirement is that Picard's theorem holds, that is, \(f\) takes all but possibly two extended complex values infinitely often in the union of any infinite subcollection of the \(C_j\). The corresponding Julia directions are obtained by angles which are accumulations of the arguments of the \(z_j\). Also, with so-called Zalcman lemma in mind, one can form a family of a certain \textit{rescaling} of the function~\(f\) corresponding to the \(C_j\) and study its normality and the value distribution of its possible non-constant limit functions. The existence of cercles de remplissage, and thus those of Julia directions, seems to have been established first by \textit{G. Valiron} [Directions de Borel des fonctions méromorphes. - Mém. Sci. Math. 89, Gauthier-Villars, Paris (1938; Zbl 0018.07303)] under the assumption \((T2)\) \[ \limsup_{r\to\infty} \frac{ T(r,f) }{ (\log r)^2 }=\infty, \] which is of course weaker than \((T3)\). Lo Yang has essentially proved that a transcendental meromorphic function \(f\) with \((T3)\) possesses a Julia direction that is a Hayman direction. As in \textit{J. Schiff}'s monograph [Normal families. Universitext. New York: Springer-Verlag (1993; Zbl 0770.30002)], Picard's theorem and Julia's theorem can be proved by Montel's approach using normal families and therefore by Marty's theorem with the spherical derivative \(f^{\sharp}:=|f'|/(1+|f|^2)\). The statement is that the existence of a sequence \(z_j\to\infty\) with the condition \((S1)\) \[ |z_j|f^{\sharp}(z_j)\to\infty \] permits \(f\) to possess a Julia direction. The points \(z_j\) serve as the centers of the \(C_j\) for~\(f\). It is natural to call the functions that do not satisfy \((S1)\) \textit{Julia exceptional functions} and to expect that they all do not have any Julia direction. The condition \((\neg S1)\) \(f^{\sharp}(z)=O(1/|z|)\) implies naturally the growth condition \((\neg T2)\) \(T(r,f)=O\bigl((\log r)^2)\). \textit{A. Ostrowski} [Math. Z. 24, No. 1, 215--258 (1926; JFM 51.0260.01)] has given an important example of Julia exceptional functions, which have no Julia directions and the Nevanlinna characteristic of which grows asymptotically like \((\log r)^2\). Hence \((T2)\) is sharp in Julia's theorem. \textit{J. Rossi} [Ann. Acad. Sci. Fenn., Ser. A I, Math. 20, No. 1, 179--185 (1995; Zbl 0817.30016)] proved by an example that \((T3)\) is sharp for the above formulation of Yang's theorem, too. In fact, for his function, the Nevanlinna characteristic grows asymptotically like \((\log r)^3/3\) and has exactly two Julia directions, but none of them is a Hayman direction. The author of the paper under review first restates a result by \textit{P. Fenton} and \textit{J. Rossi} [J. Aust. Math. Soc. 72, No. 1, 131--136 (2002; Zbl 1004.30018)] for the formulation of Yang's theorem as Theorem~4. The existence of a sequence \(z_j\to\infty\) with the condition \((S\ell)\) \[ \frac{|z_j|f^{\#}(z_j)}{\log |z_j|} \to\infty \] permits \(f\) to possess a Julia direction that is a Hayman direction. He remarks that the relation between \((S\ell)\) and \((T3)\) is unlike the relation between \((S1)\) and \((T2)\), since the condition \((\neg S\ell)\) \(f^{\sharp}(z)=O(\log|z|/|z|)\) implies only \(T(r,f)\sim(\log r)^4\). Then he defines a Hayman exceptional function \(f\) with the help of the family of functions \(f_k(z)=z^{-k}f(z)\) \((k\in\mathbb{N})\) the \(f_k^{\sharp}\) of which will satisfy \((\neg S1)\) in a sector. Theorem~9 confirms the existence of a Hayman direction of \(f\) by means of the condition \((S1)\) concerning the family \(\{f_k\}\). This is a consequence of the application of Ku's criterion to this family. With this theorem, the author finds the interesting fact that an Ostrowski's function possess a Hayman direction that is not a Julia direction, which seems to be the first example of this type. Further, by using Rossi's function, he observes that Hayman exceptional functions do not satisfy \((T2)\) in general. On the other hand, his Theorem~11 shows that every Hayman exceptional function \(f\) satisfies \((\neg T3)\) \(T(r,f)=O\bigl((\log r)^3\bigr)\). This paper is closed by Theorem~12 which answers negatively the question whether there exist transcendental meromorphic functions \(f\) such that \(f\) and all its derivatives \(f^{(k)}\) have \textit{no} Julia directions. Two problems seem to be left still unsolved: whether Yang's growth condition \((T3)\) is sharp for a Hayman direction that is not a Julia direction, and whether there exists a transcendental meromorphic function without any Hayman direction. The first one may be rephrased by Problem~1.31 in [\textit{K. F. Barth, D. A. Brannan} and \textit{W. K. Hayman}, Bull. Lond. Math. Soc. 16, 490--517 (1984; Zbl 0593.30001)], posed by D. Drasin, who noted there that one cannot expect more from Yang's method of detecting a Hayman direction through the use of filling discs. The strategy in the paper under review is an observation of the family \(\{f(z)/z^k\}_{k\in\mathbb{N}}\), in addition.