In this paper, we use purely complex analytic techniques to prove two results of the first author which were hitherto given only probabilistic proofs. A general form of the Phragmén-Lindelöf principle states that if the $p$\textsuperscript{th} Hardy norm of the conformal map from the disk to a simply connected domain is finite, then an analytic function on that domain is either bounded by its supremum on the boundary or else goes to $\ff$ along some sequence more rapidly than $e^{|z|^{p}}$. We will prove this and discuss a number of special cases. We also derive a series expansion for the Green's function of a disk, and show how it leads to an infinite product identity. The celebrated infinite product expansions for sine and cosine are realized as special cases.