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On $$L^1$$-estimates of derivatives of univalent rational functions. (English) Zbl 1395.30022
Let $$\mathcal{RU}_n$$ be the class of univalent rational functions, of degree at most $$n$$, in the unit disk $$\mathbb{D}$$, without poles in $$\overline{\mathbb{D}}$$. Let $$l(R)=\int_\mathbb{T}|R'(z)|dm(z)$$; here $$\mathbb{T}=\partial \mathbb{D}$$, $$dm(z)=|dz|/(2\pi)$$ is the normalized Lebesgue measure on $$\mathbb{T}$$, $$R\in\mathcal{RU}_n$$. The main aim of the paper is to estimate the value $$\gamma_0:=\limsup_{n\to\infty}\sup_{R\in \mathcal{RU}_n, \|R\|_{\infty,\mathbb{T}\leq 1}} \log l(R)/\log n$$ where $$\|R\|_{\infty,\mathbb{T}}=\sup\{|f(z)|: z\in \mathbb{T}\}$$. Theorem 1.2 states that $$B_b(1)\leq \gamma_0\leq 1/2$$; here $$B_b(t)$$ is the integral means spectrum for bounded univalent functions.
The authors also discuss a result by E. P. Dolzhenko [Anal. Math. 4, 247–265 (1978; Zbl 0425.41027)] on estimates of integral means for derivatives of (multivalued) bounded in $$\overline{\mathbb{D}}$$ rational functions and give a new shorter proof of the fact. At the final part of the paper, connection with Nevanlinna domains and other related topics are given as well.

##### MSC:
 30C55 General theory of univalent and multivalent functions of one complex variable 30C75 Extremal problems for conformal and quasiconformal mappings, other methods 30J99 Function theory on the disc
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