The Rudin-Koch Theorem

The Rudin-Koch Theorem concerns functions called holomorphic functions, which are complex functions that are smooth and well-behaved. It states that if such a function is bounded (doesn't grow too large) and defined on an infinite, finitely connected domain (a complex shape with a limited number of holes), then the function must be constant within that domain. Essentially, it extends the idea that, under these conditions, the function cannot vary — highlighting a kind of stability and rigidity in complex analysis where shape and boundedness impose strict limitations on the behavior of these functions.