Extremal problems for hardy classes of banach-space-valued functions and the geometry of the space of values

Let Hp (E), 1≤p≤∞, be a Hardy space of analytic functions from an open unit disk of a complex plane to a complex Banach space E. We define a class of linear operators L,L: Hp (E) → E, which are in a certain sense averages of the “boundary values”, and for a given L, study the points of maximum of norm ‖L f‖, where f lies on the unit sphere. The relationships between the geometry (strong complex convexity) of the unit sphere S:E in E, the general form of extremal functions f ∊ SHp (E), and their location on SHp (E) are established.