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Certain subclasses of analytic functions associated with the generalized hypergeometric function. (English) Zbl 1040.30003

For two holomorphic functions in the unit disk |z|<1, f(z)=z+ n=2 a n z n and g(z)=z+ n=2 b n z n the Hadamard product (or convolution ) is defined as (fg)(z)=z+ n=2 a n b n z n · Let q F s (α 1 ,...α q ;β 1 ,...,β s ;z) denote the generalized hypergeometric function with qs+1, q,s=0,1,2,..., α j , β j {0,-1,-2...}. Very special subclasses of holomorphic functions f(z)=a 1 z- n=2 a n z n , a n 0, n2, |z|<1 defined by convolution of f with z q F s and subordinate to 1+Az 1+Bz, 0B1, -BAB are studied.

(They satisfy moreover the condition f(ρ)=ρ or f ' (ρ)=1, 0<ρ<1.) The necessary and sufficient conditions in the terms of coefficients for f to be in the class under consideration are determined. Some distortion theorems and the radii of convexity and starlikeness are found.


MSC:
30C45Special classes of univalent and multivalent functions
33C20Generalized hypergeometric series, p F q