A Sneaky Proof of the Maximum Modulus Principle

A proof for the maximum modulus principle (in the unit disc) is presented. This proof is unusual in that it is based on linear algebra. The goal of this note is to provide a neat proof of the following version of the maximum modulus principle. Theorem 1 Let f be a function analytic in a neighborhood of the closed unit disc D = {z ∈ C : |z| ≤ 1}. Then max z∈D |f(z)| = max z∈∂D |f(z)|. (Here and below, ∂D denotes the unit circle ∂D = {z ∈ C : |z| = 1}). Familiar proofs derive this theorem from the open mapping principle [1, 6], from Cauchy’s integral formula via the mean value property for analytic functions [2, 3, 8], or from the maximum principle for subharmonic functions [5]. There is also a direct proof which uses the power series representation [7]. The proof I will present uses linear algebra, and is motivated by [9] and [10]. Before presenting the proof, let me review the main ingredients. For every x = (x1, . . . , xn) ∈ C, we denote ‖x‖ = √ |x1| + . . .+ |xn|. If A is an m× n matrix, we define its operator norm by ‖A‖ = sup ‖x‖=1 ‖Ax‖. The only properties of the operator norm that we will require are the following three properties, which are easy consequences of the definition. (a) ‖AB‖ ≤ ‖A‖‖B‖ for all (appropriately sized) matrices A,B. (b) If D = diag(d1, . . . , dn) then ‖D‖ = maxi |di|. (c) If A is unitarily equivalent to B, then ‖A‖ = ‖B‖.