Zeros of successive derivatives of analytic functions having a single essential singularity. II (Q1177188)

[For part I, see \textit{A. Edrei}, Analysis of one complex variable, Proc. Am. Math. Soc. 821st Meet., Laramie, Wyo. 1985, 64-98 (1987; Zbl 0641.30029).] G. Pólya defined the final set \(S(f)\) of the transcendental analytic function as the sets of all points \(z_ 0\) such that every neighborhood of \(z_ 0\) contains zeros of \(f^{(n)}(z)\) for infinitely many \(n\). The main result of the first part of this paper is Theorem 1. Let \(g(z)\) be a real entire function of finite order which is unbounded in every halfplane \(\hbox {Re }z>\sigma\) and which does not tend to \(\infty\) as \(z\to\infty\) along the positive real axis. Let \(f(z)=g(1/z)\). Then the non-negative real axis belongs to \(S(f)\). In the second part of the paper very precise results on the distribution of the real zeros of \(f^{(n)}(z)=(d/dz)^ n[z^{-1} h(z^{-1})]\) are given when \(h(z)\) is a real entire function of non-integral order \(\lambda\) with real zeros, the counting functions of the negative zeros and the positive zeros satisfy respectively \[ n^ -(r)\sim r^ \lambda,\quad n^ +(r)\sim\kappa r^ \lambda \quad (0\leq\kappa\leq 1) \qquad [\kappa=0 \hbox { means } n^ +(r)\equiv\sigma]. \] The restriction to non-integral \(\lambda\) is not essential. As an illustration asymptotic lower bounds for the number of zeros of \((d/dz)^ n(1/\Gamma(1/z))\) in any finite interval of the positive or negative real axis are given.