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

edit

Let   be a half-strip in the complex plane:

 

Suppose that   is holomorphic (i.e. analytic) on   and that there are constants  ,  , and   such that

 

and

 

Then   is bounded by   on all of  :

 

Proof

edit

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

References

edit
  • Edwards, H.M. (2001). Riemann's Zeta Function. New York, NY: Dover. ISBN 0-486-41740-9.