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Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables. (English) Zbl 1165.32006
This is a well written and interesting paper, which contains various results related to sharp growth theorems and coefficient bounds for subsets of normalized starlike mappings \(f\) on the unit ball \(B\) in a complex Banach space, such that \(z=0\) is a zero of order \(k+1\) for the mapping \(f(z)-z\). These results improve many recent results that have been obtained in the literature. The first and second sections are introductory parts, which include notions and results that are useful in the forthcoming sections. The third section is concerned with sharp growth and covering results for various subsets of the set \(S_{k+1}^*\) which consists of normalized starlike mappings on \(B\) such that \(z=0\) is a zero of order \(k+1\) for the mapping \(f(z)-z\). In the fourth section, the authors obtain a sharp estimate for the \((k+1)\)-th order coefficients of mappings in the set \(S_{g,k+1}^*(B)\). There are also obtained various particular cases of this result. The paper ends with examples that illuminate the usefulness of the above results.

MSC:
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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