Stankiewicz, Jan Subordination, convolution and some subclasses of regular functions. (English) Zbl 0862.30008 Mathematica 36(59), No. 2, 239-244 (1994). 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. Reviewer: S.Walczak (Łódź) Cited in 1 Review 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 PDFBibTeX XMLCite \textit{J. Stankiewicz}, Mathematica 36(59), No. 2, 239--244 (1994; Zbl 0862.30008)