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Some distortion and other properties associated with a family of the \(n\)-fold symmetric Koebe type functions. (English) Zbl 1259.30022

Summary: In a recent work by M. Kamali and H. M. Srivastava [JIPAM, J. Inequal. Pure Appl. Math. 5, No. 3, Paper No. 57, 8 p., (2004; Zbl 1058.30015)], a certain family of the \(n\)-fold symmetric Koebe type functions was introduced and studied systematically. In an earlier investigation, A. Eguchi and S. Owa [RIMS Kokyuroku 1192, 13–42 (2001; Zbl 0985.30506)] had considered its special case when \(n=1\). Here, in our present sequel to these earlier works, this general family of the \(n\)-fold symmetric Koebe type functions is further studied and several distortion theorems and other properties such as the radii of spirallikeness, the radii of starlikeness and the radii of convexity, which are associated with this family of \(n\)-fold symmetric Koebe type functions, are obtained. We also provide certain criteria that embed this family of functions in a function class \(\mathcal{G}_{\lambda}\) which was introduced and studied earlier by Silverman.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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