On certain subclasses of bounded univalent functions. (English) Zbl 0755.30023

Fuka, J.; Jakubowski, Z. J.

Ann. Pol. Math. 55, No. 1-3, 109-115 (1991).

Let \(\mathbb{C}\) be the complex plane, \(\mathbb{D}=\{z\in\mathbb{C}\mid\;| z|<1\}\), \(\mathbb{T}=\{z\in\mathbb{C}\mid\;| z|=1\}\), \(S=\{f\mid\;f(z)=z+a_ 2 z^ 2+\dots\) for \(z\in\mathbb{D}\), \(f\) univalent in \(\mathbb{D}\}\). Fix \(0<m\leq M<\infty\), \(0\leq\alpha\leq 1\). The subclass \(S(M,m;\alpha)\subset S\) of bounded functions \(f\) for which there exists an open arc \(I_ \alpha=I_ \alpha(f)\subset\mathbb{T}\) of length \(2\pi\alpha\) such that \(\displaystyle\overline{\lim_{{z\to z_ 0} \atop {z\in\mathbb{D}}}}| f(z)|\leq M\) for \(z_ 0\in I_ \alpha\) and \(\displaystyle\overline{\lim_{{z\to z_ 0} \atop {z\in\mathbb{D}}}}| f(z)|\leq m\) for \(z_ 0\in\mathbb{T}\setminus\overline {I}_ \alpha\) is introduced. It is shown that the condition \(M^ \alpha m^{1- \alpha}\geq 1\) is necessary for \(S(M,m;\alpha)\neq\emptyset\). A sufficient condition for \(S(M,m;\alpha)\neq\emptyset\) is given expressing analytically the fact that the Pick’s function \(p(z;M)=Mk^{- 1}[k(z)/M]\), where \(k(z)=z/(1-z)^ 2\) is the Koebe function, is lying in \(S(M,m;\alpha)\). The compactness of \(S(M,m;\alpha)\) in the topology of uniform convergence on compact subsets of \(\mathbb{D}\) is proved.

Reviewer: J.Fuka (Praha)

MSC:

30C55

General theory of univalent and multivalent functions of one complex variable

Keywords:

two constant theorem; harmonic measure; univalent; bounded functions; Pick’s function; Koebe function; compactness; topology of uniform convergence