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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, $\left(0\le k<\infty \right)$, 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 $1+z{f}^{\text{'}\text{'}}\left(z\right)/{f}^{\text{'}}\left(z\right)$, where $f\in k$-$𝒰𝒞𝒱$, is a domain bounded by the conic curves, which kind depends of the parameter $k$. The estimates of $|{f}^{\text{'}\text{'}}\left(z\right)|$, 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:
 30C45 Special classes of univalent and multivalent functions 33E05 Elliptic functions and integrals