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On new classes of analytic functions with negative coefficients. (English) Zbl 0582.30009

The author considers the class \(K^*_ n\), \(n\in {\mathbb{N}}\cup \{0\}\), consisting of functions f(z) which satisfy in the unit disc \(| z| <1\), the conditions: \[ 1)\quad f(z)=z-\sum^{\infty}_{k=2}a_ kz^ k,\quad a_ k\geq 0,\quad and \]
\[ 2)\quad Re\{D^{n+1} f(z)/D^ n f(z)\}>,\quad where\quad D^ n f(z)=f(z)*(z/(1-z)^{n+1}), \] (*) is the Hadamard product of power series. Coefficient inequalities, distortion and closure results are obtained.
Reviewer: H.S.Al-Amiri

MSC:

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