Let $f$ be a function which is analytic in a neighborhood of $0$ with $f(0)=0$ and $f'(0)=e^{2\pi i \alpha} =:\lambda$(where R\Q). The famous center problem asks when the function $f$ is linearizable near $0$. That is, does there exist a local change of coordinate $\phi$ near $0$ such that $\phi(f(z)) = \lambda \phi(z)$?

In the case where $f$ is an entire function and $0$ is linearizable,$0$ is the center of a Siegel disk, $U$; that is, a maximal domain on which $f$ is linearizable. It is an important question to determine when the boundary of $U$ contains a singular value of $f$.

Herman proved the following result: if the rotation number $\alpha$ is of diophantine type, the Siegel disk $U$ is bounded and $f\vert _{\partial U}$ is a homeomorphism, then $\partial U$ contains a critical point. While this result is rather satisfactory when $f$ is a polynomial, it is far less complete for transcendental entire functions. We will show that, if $f$ is a transcendental entire function whose set of singular values is bounded, then the assumption that $U$ is bounded can be removed. This is a considerable improvement of Herman's result in this case, and in particular answers a question of Herman, Baker and Rippon.