Proof of the Maximum Modulus Principle of an Analytic Function

Abstract

In the theory of functions of a complex variable, a well-known fact is the principle of maximum modulus of an analytic function of a complex variable, which states that if a function is analytic in a bounded domain and continuous on its boundary, is not a constant, then its modulus reaches its maximum value only at the points of the boundary. In the literature, this statement is proved rather cumbersome by contradiction by calculating the value of the function along a closed contour using the Cauchy integral formula. It would be interesting to obtain another, simpler proof of the principle of maximum modulus of an analytic function of a complex variable. This article provides a simpler, more rigorous proof of the maximum principle for the modulus of an analytic function of a complex variable. The modulus of a function of a complex variable is considered as a function of two variables. The proof is based on the calculation of partial derivatives of the first and second orders of the modulus of the function, the construction of a matrix of a quadratic form based on partial derivatives of the second order, and the analysis of the sign-definiteness of this form using the Sylvester criterion. It is proved that the value of the principal minor of the second order is less than zero inside the closed region and, therefore, the modulus of the function does not have an extremum.