Q:

Holomorphic function with constant norm on a given analytic simple closed curve

Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{C}$ which surrounds origin. 

Is there a non constant  entire holomorphic function $f$ such that $|f(z)|$ is constant on $\gamma$? 

A:

Let $\phi$ be the conformal map of the unit disk onto the interior.  The only thing that
can be said about $\phi$ is that it is analytic and univalent in the closed disk. 
If your entire function $f$ exists, then $B=f\circ\phi$ is a finite Blaschke product (by symmetry principle).  So
$\phi=f^{-1}\circ B$. It is clear that not every univalent function analytic
in the closed disk has such representation with entire $f$. For example, Iversen's theorem shows that $f^{-1}\circ B$ cannot have a singular arc. 
EDIT. 
If you want a more elementary argument, take any polynomial (univalent in the unit disk, and other than a monomial) as 
$\phi$. Then $f\circ\phi=B$, a Blaschke product.  It follows that $f$ itself is a polynomial, and thus $B(z)=z^n$. But then $\phi$ must be also a monomial, and this contradicts our assumption about $\phi$.

