## 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.)

### Keywords:

close-to-convex functions

Zbl 0311.30007