A theorem on maximum modulus

Abstract

then xC8(D), the boundary of D. If x is the only point for which the above identity holds, then x is the peak point for f in D. If x is the peak point for a bounded analytic function in D, then x is a peak point of D. We are interested in knowing which boundary points of D can be peak points. W. Rudin [I] defines a boundary point x of a domain D as a removable boundary point if every function bounded and analytic in D can be continued at x. All boundary points which are not removable are essential. We shall show that a point is a peak point of D if and only if it is an essential boundary point.