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

[alligatorman]

I'm trying to show that if $$f(x)$$ is analytic, then for large enough n,

$$|| f^{(n)} (x) || \leq c n! || f(x) ||$$,

where
$$|| f ||^2=\int_a^b{|f|^2}dx$$

and $$f^{(n)}$$ 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

 

[alligatorman]

I figure I can reduce the problem to showing that there is an N such that for all n>N, and for all $$x\in(-R,R)$$,

$$| f^{(n)} (x) | \leq c n! | f(x) |$$.


Is this necessarily true for analytic functions?