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Subordination, convolution and some subclasses of regular functions. (English) Zbl 0862.30008

Denote by \(U\) and \(H(u)\) the unit disc \(U=\{z\in\mathbb{C};|z|<1\}\) and the class \(H=H(u)\) of all regular functions in \(U\), respectively. For \(f,g\in H\), \(f(z)= a_0+a_1z+a_2z^2+\cdots\);\(g(z)=b_0+b_1z+b_2z^2+\cdots\), we define the convolution (Hadamard product) \((f*g)(z)=a_0b_0+a_1b_1z+a_2b_2z^2+\cdots\). Let \(Q\) and \(V\) be two subclasses of \(H\). By the convolution of \(Q\) and \(V\) we mean the set of functions \(Q*V=\{h=f*g; f\in Q, g\in V\}\). For given \(A\in\mathbb{C}\) and \(B\in\overline U\), we define \[ P(A,B)=\Biggl\{p\in H: p(z)\prec{1+Az\over 1-Bz}\Biggr\}, \] where “\(\prec\)” denotes the subordination. The main result of this paper is the following theorem: \(P(A,B)*P(C,D)\subseteq P(AC+AD+BC,BD)\) where \(A\), \(B\), \(C\), \(D\) are some complex numbers, \(|B|<1\) and \(|D|<1\). In addition, if \(|B|=1\) or \(|D|=1\), then the equality holds in the above inclusion.

MSC:

30C35 General theory of conformal mappings
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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