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On a subclass of close-to-convex functions with negative coefficients. (English) Zbl 0711.30014

Let T denote the class of functions f of the form \(f(z)=z- \sum^{\infty}_{n=2}| a_ n| z^ n\), analytic and univalent in the disc \(E=\{z:| z| <1\}\), \(T^*(\alpha)\), C(\(\alpha\)), \(T^{**}(\alpha)\), \(0\leq \alpha <1\), its subclasses of functions satisfying the conditions: \[ Re\{\frac{zf'(z)}{f(z)}\}>\alpha,\quad Re\{1+\frac{zf''(z)}{f'(z)}\}>\alpha,\quad Re\{f'(z)\}>\alpha, \] for \(z\in E\), respectively.
In the paper, the author obtains for functions in the class \(T^{**}(\alpha):\) sharp coefficient estimates, the distortion and covering theorems and he proves that \(C(\alpha)\subset T^{**}(\alpha)\subset T^*(\alpha)\). The classes \(T^*(\alpha)\) and C(\(\alpha\)) were considered earlier by H. Silverman [Proc. Am. Math. Soc. 51, 109-116 (1975; Zbl 0311.30007)].
Reviewer: J.Kaczmarski

MSC:

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

Citations:

Zbl 0311.30007
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