Jacob Thorstenson: The Phragmén-Lindelöf Principle

Abstract: The Phragmén-Lindelöf principle shows how holomorphic functions in unbounded regions can be bounded by their boundary values, under suitable growth conditions. In this paper we prove the maximum modulus theorem and extend it to unbounded regions using the Phragmen-Lindelöf method, to be applied in right half plane, sectors and general unbounded regions. We also include a proof of the three-lines theorem.
We then consider entire functions of exponential type. Through the indicator function, we show key properties and growth estimates. Finally we prove Carlson's theorem, showing that under suitable growth conditions, an entire function is uniquely determined by its values at the integers.