The well known Jordan-Schoenflies theorem says: let $C\subset \mathbb{R}^2$ be a closed simple curve. Then there exists a homeomorphism $f\colon\mathbb{R}^2\to \mathbb{R}^2$ such that $f(C)$ is the unit circle. Moreover if the curve $C$ is smooth then $f$ can be chosen to be diffeomorphism.

I need a slight generalization of the above statement.Let $D$ be the closed unit disk in the 2-plane. Let $C\subset D$ be a closed simple curve in the interior of $D$. Does there exist a homeomorphism $f\colon D\to D$ such that $f(C)$ is the circle of radus $1/2$ centered at 0? (For my purposes $C$ might be assumed to be a smooth curve, i.e. smooth immersion of the standard circle.) A reference would be helpful.

Remark. The point is that $f$ is a homeomorphism of the closed disk. For open one the statement is obviously equivalent to the original Jordan-Schoenflies theorem.