Phragmén–Lindelöf principle

In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function ${\displaystyle f}$ (i.e, ${\displaystyle |f(z)|<M\ \ (z\in \Omega )}$) on an unbounded domain ${\displaystyle \Omega }$ when an additional (usually mild) condition constraining the growth of ${\displaystyle |f|}$ on ${\displaystyle \Omega }$ is given. It is a generalization of the maximum modulus principle, which is only applicable to bounded domains.