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On certain classes of p-valent functions. (English) Zbl 0601.30018
The author defines the new class $$V_ k^{\lambda}(\alpha,b,p)$$ (k$$\geq 2$$, b-a non-zero complex number, $$0\leq \alpha <p$$ and $$| \lambda | <\pi /2)$$ of functions $$f(z)=z^ p+\sum^{\infty}_{n=p+1}a_ nz^ n$$ analytic in $$E=\{z\in {\mathbb{C}}:| z| <1\}$$ having (p-1) critical points in E satisfying $\limsup_{r\to 1^- }\int^{2\pi}_{0}| \frac{Re\{e^{i\lambda}[p+(1/b)(1+zA(z)-p)]- \alpha \cos \lambda}{p-\alpha}| d\theta \leq k\pi \cos \lambda,$ with $$A(z)=f''(z)/f'(z)$$, and establishes representation formula, coefficient estimates, distortion and rotation theorems for this class. He also finds the sharp radius of convexity and the Hardy class for the class $$V_ k^{\lambda}(\alpha,b)$$. These results generalize various results existing in the literature.
Reviewer: R.Parvatham

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