## 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.

### MSC:

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

### Keywords:

negative coefficients