Functions in N + with the positive real parts on the boundary, and extremal problems in H1

An essentially bounded function φ on the unit circle gives a continuous linear functional T φ on the Hardy space H 1 ρ(φ) denotes a set of all complex numbers s such that there exists at least one function which attains the norm of T φ− s In a previous paper, we showed that is empty or an open disc.Unfortunately we did not know when ρ(φ) is open or closed. In this paper, we study when ρ(φ) is open or closed. Moreover the functions in the Smirnov class N + whose real parts are nonnegative on the unit circle are described and studied. Then we give new characterizations of exposed points in the unit ball of H 1 and we determine when the sum of two inner functions is outer. As an result, we can describe all functions which have their Denjoy-Wolff points on the unit circle.