Holomorphic functions with positive real part

The main purpose of this note is to prove a special case of the following conjecture. Conjecture. If F is holomorphic on the unit ball B n in C n and has positive real part, then F is in H p (B n ) for 0 < p < ½( n + 1). Here H p (B n ) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on B n . See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [ 1 , Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functions is in H p ( B 2 ) for 0 < p < 3/2.