On univalent functions with negative coefficients. (English) Zbl 0766.30010

Let \(H(U)\) denote the set of all functions \(f\) holomorphic in the unit disc \(U\) and \(D^ n\), \(n\in\mathbb{N}=\{0,1,2,\dots\}\) be the operator such that \(D^ n: H(U)\to H(U)\) and \(D^ 0 f(z)=f(z)\), \(D^ 1 f(z)=zf'(z)\), \(D^ n f(z)=D(D^{n-1} f(z))\).
The author considers the class \(T_ n(\alpha,\beta)\), \(\alpha\in\langle 0,1)\), \(\beta\in(0,1\rangle\), \(n\in\mathbb{N}\), of functions \(f\) in \(H(U)\) of the form \[ f(z)=z-\sum_{k=1}^ \infty a_ k z^ k, \qquad a_ k\geq 0, \qquad k=2,3,\dots \] and such that \(| J_ n(f,\alpha;z)|<\beta\) where \[ J_ n(f,\alpha;z)=\left[ {{D^{n+1}f(z)} \over {D^ n f(z)}}-1\right] \left/\left[ {{D^{n+1}f(z)} \over {D^ n f(z)}}+1-2\alpha\right]\right.,\qquad z\in U. \] He obtains many properties of the considered class.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)