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Rigidity and Parametric Embedding of Semi-complete Vector Fields on the Unit Disk

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Abstract

This manuscript contains recent results on generation theory of semigroups of holomorphic mappings with applications to geometric function theory as well as new results which could define some new trends in the development of the subject. We present various characterizations, properties and methods of parametric embedding of the class of semi-complete vector fields (holomorphic generators) and their relations to the classes of starlike and spirallike functions on the unit disk. In particular, we show that the class of univalent functions satisfying the Nashiro–Warshawskii condition consists of semi-complete vector fields.

In a parallel manner, we present certain quantitative characteristics of boundary regular null points of semi-complete vector fields and corresponding backward flow invariant domains. In addition, we establish generalized infinitesimal versions of the Burns-Krantz rigidity theorem. Some open questions are discussed.

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  1. Department of Mathematics, ORT Braude College, 21982, Karmiel, Israel

    David Shoikhet

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Shoikhet, D. Rigidity and Parametric Embedding of Semi-complete Vector Fields on the Unit Disk. Milan J. Math. 84, 159–202 (2016). https://doi.org/10.1007/s00032-016-0254-5

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