Holomorphic functions with positive real part on the unit ball of \({\mathbb{C}}^ n\) (Q1115991)

The set \({\mathcal P}\) of holomorphic functions on the open unit ball B of \({\mathbb{C}}^ n\) which have positive real part and take the value 1 at 0 is studied. The main result is the following: Suppose that p is a homogeneous polynomial which is an extreme point of the closed unit ball of the space \(H^{\infty}(B)\), then there is a polynomial q of degree less than the degree of p such that the function \((1+p+q)/(1-p)\) is an extreme point of \({\mathcal P}\). Some examples are also considered.