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Conic regions and k-uniform convexity. II. (English) Zbl 0995.30013

[For part I see the the authors in J. Comput. Appl. Math. 105, No. 1-2, 327-336 (1999; Zbl 0944.30008).]

Summary: In the previous paper, due to the authors, the class of k-uniformly convex functions, (0k<), has been introduced. The mentioned class, denoted by k-𝒰𝒞𝒱, is a generalization of the class of uniformly convex functions introduced by Goodman, and studied by Rønning, Ma and Minda. This paper is a continuation of the investigation of the class k-𝒰𝒞𝒱, for that the region of values of 1+zf '' (z)/f ' (z), where fk-𝒰𝒞𝒱, is a domain bounded by the conic curves, which kind depends of the parameter k. The estimates of |f '' (z)|, bounds of coefficients of k-uniformly convex functions, the sharp bounds on |a n |, for n=2,3,4, and the extremal functions which realize equality are given. Some particular examples of functions having the required properties are found.

MSC:
30C45Special classes of univalent and multivalent functions
33E05Elliptic functions and integrals