11 M. Elin The Galilee Research Center for Applied Mathematics of ORT Braude College Karmiel, Israel.

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Presentation transcript:

11 M. Elin The Galilee Research Center for Applied Mathematics of ORT Braude College Karmiel, Israel

Continuous Semigroups

33 Spirallike and starlike mappings

44 In the one-dimensional settings - the well-known criteria of Nevanlinna, Study and Špaček In multi-dimensional situations – Suffridge, Gurganus, Pfaltzgraff, Gong, … In multi-dimensional situations – not all of the analogues hold, proofs are very complicated, examples are rather hard to construct. Spirallike and starlike mappings

55 Since the work of Roper and Suffridge in 1995, there has been considerable interest in constructing holomorphic mappings of the unit ball in a Banach space with various geometric properties by using mappings with similar properties acting in a subspace. Such properties include convexity, starlikeness, spirallikeness, and so on. It is also of interest to extend subordination chains, semigroups and semigroup generators. Extension Operators

Roper-Suffridge extension operator

7 Modifications of R-S extension operator Pfaltzgraff, Suffridge, 1997 I.Graham, G.Kohr, M.Kohr, 2000 I.Graham, G.Kohr, 2000

8 Modifications of R-S extension operator - the chain rule is invertible

99 Some notations

10 Main notation and notion

11 Extension operators for semigroups

12 Extensions of spirallike mappings and subordination chains

13 Extensions of spirallike mappings

14 Extensions of spirallike mappings

15 Extensions of spirallike mappings

16 Extension operator: Extreme Points, Support Points Further question

Muir’s extension operator

Geometric explanation

Covering results

Roper-Suffridge type operator

Spiralikeness for Muir’s type operator

23 Thank you for your attention!