Entire functions with prescribed singular values

Abstract

We introduce a new class of entire functions [Formula: see text] which consists of all [Formula: see text] for which there exists a sequence [Formula: see text] and a sequence [Formula: see text] satisfying [Formula: see text] for all [Formula: see text]. This new class is closed under the composition and it is dense in the space of all nonvanishing entire functions. We prove that every closed set [Formula: see text] containing the origin and at least one more point is the set of singular values of some locally univalent function in [Formula: see text], hence, this new class has nontrivial intersection with both the Speiser class and the Eremenko–Lyubich class of entire functions. As a consequence, we provide a new proof of an old result by Heins which states that every closed set [Formula: see text] is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally, we show that the class [Formula: see text] contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.