On a class of functions close to functions of bounded boundary rotation. (English) Zbl 0704.30017

Gopalakrishna, H. S.; Umarani, Prakash G.

Tamkang J. Math. 20, No. 4, 303-310 (1989).

Denoting the class of all regular functions f(z) in the open unit disc with \(f(0)=0\), \(f'(0)=1\) by A and the class of functions of bounded boundary rotation by \(V_ k\) the authors define the class W(k,\(\alpha\)) for \(k\geq 2\) and \(| \alpha | \leq \pi /2\) as follows. f in A belongs to W(k,\(\alpha\)) if and only if \[ Re\{e^{i\alpha}(\frac{f'(z)}{g'(z)})\}>0\text{ for } | z| <1 \] for some \(g\in V_ k\). Let \(W_ k=\cup_{\alpha}W(k,\alpha)\). Sharp radii of convexity and close-to-convexity, distortion theorems bounds for \(f'(z)\) are obtained using Goluzin’s method of variations for the classes W(k,\(\alpha\)) and/or W(k).

Reviewer: V.Karunakaran

MSC:

30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C70	Extremal problems for conformal and quasiconformal mappings, variational methods

Keywords:

radius of close-to-convexity; bounded boundary rotation; radii of convexity