Sorry I don't know how to give an appropriate title.

In the complex plane, suppose there is a graph $x+if(x)$ separating the plane into two unbounded components, where $f(x)$ is smooth and bounded, with all derivatives bounded. $f(x)$ could oscillate at $\infty$. Let $\Omega$ denote the region under the graph.

By Riemann mapping theorem, there is a Riemann mapping $\Phi: \Omega\rightarrow \mathbb{P}_-$, where $\mathbb{P}_-$ is the lower half plane.

My question is, is it possible to find a Riemann mapping $\Phi$ such that $\Phi$ grows almost like the identity, in the sense that $\Phi(z)-z$ is bounded?