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Neighborhoods of certain subclasses of analytic functions of complex order with negative coefficients. (English) Zbl 1140.30005

Let \(\mathcal{A}(n),\) \(n=1,2,\ldots,\) be the class of functions that are analytic in the open unit disc of the form \(f(z)=z-\sum_{k=n+1}^\infty a_kz^k,\) \(a_k\geq0.\) The \((n,\delta)\)-neighborhood of the function \(f(z)\) is defined by
\[ N_{n,\delta}(f) = \left\{ g: g\in \mathcal{A}(n),\;g(z)=z-\sum_{k=n+1}^\infty b_kz^k,\;\sum_{k=n+1}^\infty k| a_k-b_k| \leq\delta\right\}. \]
In this paper the authors investigate the \((n,\delta)\)-neighborhoods of several subclasses of \(\mathcal{A}(n)\) of normalized analytic functions with negative and missing coefficients which are introduced by use of the Ruscheweyh derivative.

MSC:

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