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$$\Phi$$-like and convex mappings in infinite dimensional spaces. (English) Zbl 1092.32501
Summary: Let $$\mathbb B$$ be the unit ball in a complex Banach space. Gurganus gave a sufficient condition under which a normalized locally biholomorphic mapping $$f$$ on $$\mathbb B$$ is univalent on $$\mathbb B$$ and $$f(\mathbb B)$$ is a $$\Phi$$-like domain. However, we think that there is a gap in his proof. We give a complete proof of the above result. When $$\mathbb B$$ is the unit ball in a complex Hilbert space, we give analytic characterizations for a locally biholomorphic mapping on $$\mathbb B$$ to be a convex mapping and also give a distortion theorem of normalized convex mappings on $$\mathbb B$$.

##### 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.) 46G20 Infinite-dimensional holomorphy 58C10 Holomorphic maps on manifolds