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p-valent classes related to convex functions of complex order. (English) Zbl 0587.30021
Let C(b,p) (b\(\neq 0\) complex, \(p\geq 1\) integer) denote the class of functions \(f(z)=z^ p+a_{p+1}z^{p+1}+...\). which are regular in the unit disc U and such that \[ Re\{p+(1/b)(1-p+zf'(z)/f(z))\}>0\quad for\quad z\in U. \] In this paper the representation formula \[ f(z)=pz^{p-1} \exp \{-2bp\int^{2p}_{0}\log (1-ze^{-it})d\mu (t) \] and the estimations of the modulus of the derivative and the coefficients for the functions of the class C(b,p) are given. In the special cases C(1,1), C(b,1), C(1,p) we obtain the known subclasses of regular functions.
Reviewer: J.Stankiewicz

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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