In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip. The result is useful in the study of the Riemann zeta function, and is a special case of the Phragmén–Lindelöf principle. Also, see Hadamard three-lines theorem.

Statement of the theorem

Let \Omega be a half-strip in the complex plane: Suppose that f is holomorphic (i.e. analytic) on \Omega and that there are constants M, A, and B such that and Then f is bounded by M on all of \Omega:

Proof

Fix a point inside \Omega. Choose, an integer N>A and y_1>\tau large enough such that . Applying maximum modulus principle to the function and the rectangular area we obtain, that is,. Letting yields as required.