Angular derivatives at boundary fixed points for self-maps of the disk

Let phi be a one-to-one analytic function of the unit disk D into itself, with phi(0) = 0. The origin is an attracting fixed point for phi, if phi is not a rotation. In addition, there can be fixed points on partial derivative D where phi has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of phi is a one-to-one analytic function sigma defined on D such that phi = sigma(-1) (lambda sigma), where lambda = phi'(0). If phi(K) is the first iterate of phi that does have b.r.f.p., we compute he Hardy number of sigma, h(sigma) = sup{p > 0 : sigma epsilon H-p (D)}, in terms of the smallest angular derivative of phi(K) at its b.r.f.p.. In the case when no iterate of phi has b.r.f.p., then sigma epsilon boolean AND(p< infinity) H-p, and vice versa. This has applications to composition operators, since sigma is a formal eigenfunction of the operator C-phi(f) = f o phi. When C-phi acts on H-2(D), by a result of C. Cowen and B. MacCluer, the spectrum of C-phi is determined by lambda and the essential spectral radius of C-phi, r(e) (C-phi). Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, r(e)(C-phi) can be computed in terms of h(sigma). Hence, our result implies that the spectrum of C-phi is determined by the derivative of phi at the fixed point 0 epsilon D and the angular derivatives at b.r.f.p. of phi or some iterate of phi.