Thread Rating:
  • 1 Vote(s) - 5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Universal uniqueness criterion?
#17
Ok, I give it another try:

Proposition.
Let , , and being a domain (open and connected) of definition and being a domain (open and connected) of values for a holomorphic function.
Let be a holomorphic function on , , such that
(0) .
(1)
(2)
(U) has bounded real part.
Then for every other on holomorphic satisfying (0), (1), (2), (U).

Proof.
has bounded real part on . We consider and and so to be holomorphic on the same Riemann surface . is a 1-periodic function, holomorphic on . As it can be continued to an entire function, so it has to take on every complex value with at most one exception already on the strip otherwise it is a constant. Now has bounded real part and hence can not take on every value, so and .
Reply


Messages In This Thread
Universal uniqueness criterion? - by bo198214 - 05/21/2008, 06:24 PM
RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 05:19 AM
RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 06:42 AM
RE: Universal uniqueness criterion? - by bo198214 - 05/22/2008, 11:25 AM
RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 03:11 PM
RE: Universal uniqueness criterion? - by bo198214 - 05/22/2008, 05:55 PM
RE: Universal uniqueness criterion? - by bo198214 - 05/23/2008, 12:07 PM
Uniqueness of analytic tetration - by Kouznetsov - 09/30/2008, 07:58 AM
RE: Universal uniqueness criterion? - by bo198214 - 10/04/2008, 11:19 PM
RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 02:51 PM
RE: miner error found in paper - by bo198214 - 06/19/2009, 04:53 PM
RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 06:25 PM
RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 07:59 PM
RE: Universal uniqueness criterion? - by bo198214 - 06/20/2009, 02:10 PM
RE: Universal uniqueness criterion? - by bo198214 - 07/05/2009, 06:54 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Semi-exp and the geometric derivative. A criterion. tommy1729 0 1,101 09/19/2017, 09:45 PM
Last Post: tommy1729
  A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 9,789 02/17/2017, 05:21 AM
Last Post: JmsNxn
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 13,267 01/09/2017, 02:41 AM
Last Post: Gottfried
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,360 03/19/2016, 10:44 AM
Last Post: fivexthethird
  [2014] Uniqueness of periodic superfunction tommy1729 0 1,737 11/09/2014, 10:20 PM
Last Post: tommy1729
  Real-analytic tetration uniqueness criterion? mike3 25 19,218 06/15/2014, 10:17 PM
Last Post: tommy1729
  exp^[1/2](x) uniqueness from 2sinh ? tommy1729 1 2,009 06/03/2014, 09:58 PM
Last Post: tommy1729
  Uniqueness Criterion for Tetration jaydfox 9 10,399 05/01/2014, 10:21 PM
Last Post: tommy1729
  Uniqueness of Ansus' extended sum superfunction bo198214 4 6,188 10/25/2013, 11:27 PM
Last Post: tommy1729
  A question concerning uniqueness JmsNxn 3 5,135 10/06/2011, 04:32 AM
Last Post: sheldonison



Users browsing this thread: 1 Guest(s)