Schwarz lemma

Let be the open unit disk in the complex plane $ℂ$. Let $f:\mathrm{\Delta }\to \mathrm{\Delta }$ be a holomorphic function with $f(0)=0$.Then $|f(z)|\le |z|$ for all $z\in \mathrm{\Delta }$, and $|f^{\prime }(0)|\le 1$. If the equality $|f(z)|=|z|$ holds for any $z\ne 0$ 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 result exists for real functions, of course.