Classes of Holomorphic Functions of Several Variables in Circular Domains.

where ax depends only on X and nothing else. Also, for r1, r2 < 1, fo' ff(re219) f(r2e2tri) x dO -* 0 as (ri, r2) --(1, 1), and there exists a measurable function F(O) on the boundary r = e2'i0 such that, for 0 < r < 1, fol If(re2i0) F(O)lx do -0 as r -* 1. The quantity M(r) = fOIif(re ) | do is monotonely increasing in r. Using this, it is not hard to verify that Theorem 1 gives rise to the following generalization of itself. THEOREM 2. If our holomorphic function depends on a parameter 4, f(z) = 0 sup fx dt fol | p(re2wie; 4)|x do Cx < 00, O<< 1