On a Result of Bergweiler and Eremenko Concerning Derivatives Bounded on Preimages of Unbounded Sets

Abstract

We show that the \(k\)th derivative of a transcendental entire function \(f\) is unbounded on the preimages of unbounded sets provided that there is a value \(w^*\in \mathbb{C }\) such that \(f^{\prime },\dots ,f^{(k-1)}\) are bounded on the fibre \(f^{-1}(\left\{ w^*\right\} )\). In particular, this holds if the spherical derivative of \(f\) is bounded.