ABSTRACT
This chapter discusses the local and boundary maxima, the maximum modulus principle, and the boundary maximum modulus theorem. Since the domain is bounded, the maximum must occur somewhere; and it cannot occur in the interior by the formulation of the maximum principle. So it must be in the boundary. Holomorphic functions (or, more precisely, their moduli) can have interior minima. It should be noted that the maximum modulus theorem is not always true on an unbounded domain. The Schwarz Lemma treats certain estimates that must be satisfied by bounded holomorphic functions on the unit disc. The chapter presents the classical, analytic viewpoint for examining the Schwarz Lemma.
