Local extremal problems for bounded analytic functions without zeros

In the class B(t), t > 0, of all functions f (z, t) = e(-t) + c(1)(t)z + c(2) (t) z(2) +... that are analytic in the unit disc U and such that 0<\f(z,t)\<1 in U, we obtain asymptotic estimates for the coefficients for small and sufficiently large t > 0. We suggest an algorithm for determining those t > 0 for which the canonical functions provide the local maximum of Re c(n) (t) in B (t). We describe the set of functionals L(f) = Sigma(n)(k=0) lambda(k)c(k) for which the canonical functions provide the maximum of Re L (f) in B (t) for small and large values of t. The proofs are based on optimization methods for solutions of control systems of differential equations.