×

zbMATH — the first resource for mathematics

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.)
PDF BibTeX XML Cite
Full Text: DOI EuDML