Generalized convolutions and classes of functions defined by subordination

Denote by HH the class of functions analytic in the unit disc. For two functions f(z)=∑n=0∞anzn and g(z)=∑n=0∞bnzn of the class HH and for given complex mm we define the generalized convolution of the form (f⋆mg)(z)=a0b0+m∑n=1∞anbnzn. By PiPi we denote the classes of functions f∈Hf∈H subordinated to the linear fractional transformations (αi+βiz)/(1−γiz)(αi+βiz)/(1−γiz), where βi+αiγi≠0,|γi|≤1βi+αiγi≠0,|γi|≤1 and i=1,2,3i=1,2,3. The purpose of this paper is to find the necessary and sufficient conditions for the inclusion P1⊂P2⋆mP3P1⊂P2⋆mP3, where P2⋆mP3={f⋆mg:f∈P2,g∈P3}. As a consequence of this result we obtain, there exist bounded analytic functions, which cannot be represented as the generalized convolution of two bounded analytic functions.