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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.

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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