Julia directions of meromorphic functions and their derivatives (Q1849624)

Let \(f\) be a transcendental meromorphic function in the complex plane \(\mathbb C\). A ray \(J\) connecting \(0\) and \(\infty\) is called a Julia direction of \(f\) if in every open sector containing \(J\) the function \(f\) takes on every value of the sphere \(\widehat{\mathbb C}\) with at most two exceptions infinitely often. A classical result of Julia states that every \(f\) with an asymptotic value has a Julia direction. This is no longer true under the weaker assumption that \(f\) is bounded on a path to \(\infty\). The main result of this article states that if a transcendental meromorphic function \(f\) is bounded on a path to \(\infty\) and has no Julia direction, then there is a common Julia direction for all derivatives of \(f\). In fact, somewhat more is shown. There are also related results under the assumption that \(f\) is \(o\bigl(\sqrt{|z|} \bigr)\) or \(O\bigl(\sqrt{|z|} \bigr)\) on a path to \(\infty\). The method of proof uses the theory of normal families. Finally, a conjecture of \textit{G.~Frank} and \textit{Y.~Wang} [N.Z.J. Math. 27, 199-205 (1998; Zbl 0960.30024)] that every Julia direction of a meromorphic function \(f\) of positive order is a Julia direction of \(f'\) is disproved.