Analytic functions that take values in a convex region

Abstract

that are convergent in E, the unit circle I zI 0 in E. We let 5(1, H) denote the family of such functions. In this notation 1=f(O), and H denotes the half-plane Tw> 0, in which the values of f(z) are required to lie for z in E. Herglotz [3] and later others (see Nevanlinna [6, p. 185]) proved that if f(z) G F(1, H) then it has a representation as a Stieltjes integral, i.e.