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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, $(0\le k<\infty)$, has been introduced. The mentioned class, denoted by $k$-${\cal U}{\cal C}{\cal V}$, 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 $k$-${\cal U}{\cal C}{\cal V}$, for that the region of values of $1+zf''(z)/f'(z)$, where $f\in k$-${\cal U}{\cal C}{\cal V}$, 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.

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