Let $\Delta=\{z:|z|<1\}$ be the open unit disk in the complex plane $\mathbb{C}$. Let $f\colon\Delta\to\Delta$ be a holomorphic function with $f(0)=0$. Then $|f(z)|\leq|z|$ for all $z\in\Delta$, and $|f^{{\prime}}(0)|\leq 1$. If the equality $|f(z)|=|z|$ holds for any $z\neq 0$ or $|f^{{\prime}}(0)|=1$, then $f$ is a rotation: $f(z)=az$ with $|a|=1$.

This lemma is less celebrated than the bigger guns (such as the Riemann mapping theorem, which it helps prove); however, it is one of the simplest results capturing the “rigidity” of holomorphic functions. No similar result exists for real functions, of course.