Radial Variation of Bounded Analytic Functions on the Disc

摘要

Let F be a bounded analytic function on the disc D = (z a member of C such thatabsolute value of z < 1). Following (Ru), define W(F,r,theta) = integral from 0 to r of absolute value of (F' (rho e(sup i(theta))))d(rho)(r < 1); V(F,theta) = W(F,1,theta). The quantity V(F,theta) corresponds to the variation of F on the radius of D terminating at the point e(sup i(theta)). It is shown in (R) that the set (theta(verticle bar)V(F,theta) < infinity) may be of measure zero (and of first category). In fact, as shown in (R), F may be taken to be a Blaschke product or an element of the disc algebra (i.e. continuous up to the boundary). The problem left open in (R) is whether V(F,theta) may be infinite in any direction. The purpose of the note is to disprove this.