**Entire functions with prescribed singular values**

*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 ${\mathcal{E}}$ which consists of all ${F}_{0}\in {\mathcal{O}}(\u2102)$ for which there exists a sequence $({F}_{n})\in {\mathcal{O}}(\u2102)$ and a sequence $({\lambda}_{n})\in \u2102$ satisfying ${F}_{n}(z)={\lambda}_{n+1}{e}^{{F}_{n+1}(z)}$ for all $n\ge 0$. 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 $V\subset \u2102$ containing the origin and at least one more point is the set of singular values of some locally univalent function in ${\mathcal{E}}$, 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 $V\subset \u2102$ 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 ${\mathcal{E}}$ contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.