Iterative characterizations of powers and exponentials
1. The only non-constant entire function satisfying the functional equation \[ F(z+F(z))=F(z)F(z^{k-1}+1), \] where \(k\geq 3\) is a fixed integer, is \(F(z)=z^ k.\) 2. The only non-constant entire functions satisfying the functional equation \[ F(z+F(z))=F'(F(z)F'(z)) \] are \(a^{-2} \exp (az)\), where a is an arbitrary non-zero complex constant. The second result seems harder to prove and depends on an application of Nevanlinna theory.
scientific article; zbMATH DE number 4095836