In a comment on question 110345 I made a claim that might be incorrect. I claimed that if
f(z) is a non-constant analytic function defined by a power series whose circle of convergence C
has a positive radius, then f(z) cannot be bounded at all points in the interior of C. But is 
this really true? Or am I just imagining that I learned it somewhere. I could not come up with
any simple counter-examples. It sounds like some weird generalization of Liouville's theorem.



Is this a "folk theorem" about analytic functions of a complex variable?
This is not true. For a counterexample take 


$$\sum\_{n=1}^{\infty} \frac{x^n}{n^2} $$


The radius of convergence is one and this is bounded by 


$$\sum\_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6 $$


You are likely confusing this with the [maximum modulus principle](http://en.wikipedia.org/wiki/Maximum_modulus_principle) saying that there must be no local maximu (in absolute value) in the interior if the function is nonconstant.


