p-valent classes related to convex functions of complex order (Q1072674)

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.