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Eventual boundedness of nth derivative of an analytic function in L2 norm

  1. Feb 24, 2009 #1
    I'm trying to show that if [tex]f(x)[/tex] is analytic, then for large enough n,

    [tex]|| f^{(n)} (x) || \leq c n! || f(x) ||[/tex],

    where
    [tex]|| f ||^2=\int_a^b{|f|^2}dx[/tex]

    and [tex]f^{(n)}[/tex] denotes the nth derivative.

    I tried to use the Taylor series, and then manipulated some inequalities, but I wasn't getting anywhere.

    Any ideas?

    Thanks
     
  2. jcsd
  3. Feb 24, 2009 #2
    I figure I can reduce the problem to showing that there is an N such that for all n>N, and for all [tex]x\in(-R,R)[/tex],

    [tex]
    | f^{(n)} (x) | \leq c n! | f(x) |
    [/tex].


    Is this necessarily true for analytic functions?
     
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