2019/08/16 by Thaler, Luka Boc
#30D05 #30D15 #30D20 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1908.06026
We introduce a new class of entire functions E which consists of all F0\inO(ℂ) for which there exists a sequence (Fn)∈ O(ℂ) and a sequence (λn)∈ℂ satisfying Fn(z)=λn+1e^Fn+1(z) for all n≥ 0. This new class is closed under the composition and its is dense in the space of all non-vanishing 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 non-trivial 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 non-empty.