The Phragmén-Lindelöf Principle and Its Applications

This thesis discusses an extension of the classical maximum modulus principle to unbounded domains, called the Phragmén-Lindelöf principle. The principle is illustrated in strips, half planes and sectors, as well as through the related Hadamard theorems. The thesis is concluded by proving applications of the Phragmén-Lindelöf principle within Fourier analysis and operator theory. The applications within Fourier analysis are the Hausdorff-Young inequality and Hardy's uncertainty principle, and the application within operator theory is the Riesz-Thorin interpolation theorem.