The requirements of this problem specialize the hypotheses of the Three Circles Theorem in two directions. First, since the functions f(z) of class W are analytic for I zI R, they are single-valued; the Three Circles Theorem admits to competition functions whose moduli are single-valued, but which need not themselves be single-valued. Second, the functions of class ?I are analytic for IzI <R, whereas the functions admitted in the Hadamard theorem are required to be analytic only for r < I z <R. Since the extremal functions of the Three Circles Theorem are analytic throughout I z I <R only when m/M is a positive integral power of r/R, the appraisal given by this theorem for M(f, p) with fE Ct is the best possible only under very restrictive hypotheses on m, M, r, R. By hypotheses II and III on fEE9, the class !R is compact and hence there certainly exists a function fo(z) of this class for which