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M. Elin

Extension operators in Geometric Function Theory. M. Elin. The Galilee Research Center for Applied Mathematics of ORT Braude College Karmiel , Israel. 1. Continuous Semigroups. Spirallike and starlike mappings. 3. Spirallike and starlike mappings.

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M. Elin

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  1. Extension operators in Geometric Function Theory M. Elin The Galilee Research Center for Applied Mathematics of ORT Braude College Karmiel, Israel 1

  2. Continuous Semigroups

  3. Spirallike and starlike mappings 3

  4. Spirallike and starlike mappings 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. 4

  5. Extension Operators 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.   5

  6. 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

  9. Some notations 9

  10. Main notation and notion 10

  11. Extension operators for semigroups 11

  12. Extensions of spirallike mappings and subordination chains 12

  13. Extensions of spirallike mappings 13

  14. Extensions of spirallike mappings 14

  15. Extensions of spirallike mappings 15

  16. Extreme Points, Support Points Further question Extension operator:

  17. Muir’s extension operator

  18. Geometric explanation

  19. Geometric explanation

  20. Covering results

  21. Roper-Suffridge type operator

  22. Spiralikeness for Muir’s type operator

  23. Thank you for your attention! 23

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