 Solution

The complex Liouville theorem states that a bounded entire function is constant. Indeed, if $|f|\le M$, the Cauchy estimate on any disc of radius $R$ centered at $z$ gives $|f^{\prime }(z)|\le M/R$. Letting $R\to \infty$ gives $f^{\prime }(z)=0$ everywhere.

The analogous conclusion for bounded real analytic functions on $\mathbb{R}$ is false. For example, $\sin x$ is bounded, nonconstant, and real analytic on the whole real line. Boundedness only on that line does not bound its holomorphic extension on the complex plane.