The main purpose of this note is to prove a special case of the following conjecture.
Conjecture.If Fis holomorphic on the unit ball Bnin C nand has positive real part, then Fis in Hp(Bn)for 0 < p< ½( n+ 1).
Hp(Bn)(0 < p< ∞) denote the usual Hardy spaces of holomorphic functions on Bn.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
Hp( B2) for 0 < p< 3/2.