Analytic Functions Whose Boundary Values have Lipschitz Modulus

Abstract

The properties of the canonical factorization of functions analytic in a disk (see [1], pp. 111–114; we employ the terminology of [the Russian edition of] Hoffman’s book [2], pp. 91–104) whose boundary values have a certain smoothness are not yet fully understood. The results of [3, 4] give sound reason to suspect that certain important classes of analytic functions smooth at the boundary are invariant under division by an internal factor. It is logical in this connection to attempt to establish a relationship between the smoothness of an external analytic function f and the smoothness of its modulus | f | on a circle; if the degree of smoothness of f and | f | were identical, the above-mentioned invariance would be quite simple to prove. The situation, however, is far more complex. We now formulate a theorem which shows, in particular, that an external function f behaves with respect to smoothness, in general, twice (but not more than twice) as badly as the function | f |, at least whenever the word “smoothness” connotes membership in a Lipschitz class of order less than unity.