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Quasiconformal and quasiregular harmonic analogues of Koebe’s theorem and applications. (English) Zbl 1221.30063

Let \(f\) be a holomorphic univalent function in the unit disk \(D\), \(f(0)=0\), \(f'(0)=1\). According to Koebe’s one-quarter theorem, \(f(D) \supset \frac 14 D\). Let now \(f\) be a quasiconformal mapping in \(D\) with complex dilatation \(\mu = \frac{ f_{\overline{z}}}{f_z}\) and let \(\mu^+(z) = \text{ess} \sup \{| \mu(\zeta)| : | \zeta| =| z| \}\). The author introduces \(\tau_f = \int_0^1 \frac{2\mu^+(t)dt}{t(1+\mu^+(t))}\) and \(\delta_f = \text{dist} (f(0), \partial f(D))\), and proves the inequality \(| f'(0)| \leq 4 \delta_f e^{\tau_f}\) for \(f\) conformal at zero and \(f(0)=0\). The author discusses Koebe’s theorem for quasiregular harmonic functions, and its applications and relations to Grötzsch’s theorem.

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C62 Quasiconformal mappings in the complex plane
30C55 General theory of univalent and multivalent functions of one complex variable
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