International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-08-05 , DOI: 10.1142/s0129167x20500755 Luka Boc Thaler
We introduce a new class of entire functions which consists of all for which there exists a sequence and a sequence satisfying for all . 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 containing the origin and at least one more point is the set of singular values of some locally univalent function in , 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 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 contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.