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 $ℰ$ which consists of all $F0∈𝒪(ℂ)$ for which there exists a sequence $(Fn)∈𝒪(ℂ)$ and a sequence $(λn)∈ℂ$ satisfying $Fn(z)=λn+1eFn+1(z)$ for all $n≥0$. 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 $V⊂ℂ$ containing the origin and at least one more point is the set of singular values of some locally univalent function in $ℰ$, 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 $V⊂ℂ$ 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 $ℰ$ contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.

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