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Bounded Montel univalent functions. (English) Zbl 0674.30011

Let a be fixed, \(0<a<1\), and let f be regular and univalent in \(\Delta =\{z:\) \(| z| <1\}\). The function f is in Montel’s class M(a) if \(f(0)=0\), \(f(a)=a\). The subclass of M(a) whose members F are bounded by M (\(| F(z)| \leq M\) for \(z\in \Delta)\) is denoted by M(a;M). In the paper, some properties of functions in M(a;M) are established. For example, if \(F(z)=A_ 1z+A_ 2z^ 2+...\), \(F\in M(a;M)\), the authors obtain sharp upper and lower bounds for \(| A_ 1|\). They also obtain two covering theorems.
Reviewer: Z.Jakubowski

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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