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

  • #1
112
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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
 

Answers and Replies

  • #2
112
0
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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