On a radius problem concerning a class of close-to-convex functions

Theorem 1. Let L be a continuous functional on B, and W0 ∈B such that (9) Re L(W0) = max W ∈B Re L(W). Assume that L has a Fr´echet derivative at W0 relative to B, and that this deriva- tive does not vanish identically over H(D). Then W0 is a finite Blaschke product. This result is important to us because the functional L defined in (5) admits, at any W0 ∈B, the Fr´echet derivative LW0(h) = rR 0 (1 −ρ2) t−r 1−rtth(t) 1 + ρ¯ξ t−r 1−rtW0(t)