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

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
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