A Herglotz-type representation for vector-valued holomorphic mappings on the unit ball of CnCn

The generalization of the Carathéodory class, those analytic functions on the open unit disk having positive real part and taking the value 1 at the origin, to the open unit ball BB of CnCn is the family MM of all holomorphic mappings f:B→Cnf:B→Cn such that f(0)=0f(0)=0, Df(0)=IDf(0)=I, and Re〈f(z),z〉>0Re〈f(z),z〉>0 for all z∈B∖{0}z∈B∖{0}, where Df is the Fréchet derivative of f, I   is the identity operator on CnCn, and 〈⋅,⋅〉〈⋅,⋅〉 is the Hermitian inner product in CnCn. We present an integral representation for functions in the class MM in terms of probability measures on the unit sphere S=∂BS=∂B similar to the well-known Herglotz representation of the Carathéodory class. This representation follows, in part, from a new integral formula of Cauchy type that reproduces a continuous f:B‾→Cn whose restriction to BB is holomorphic by using a fixed vector-valued kernel and the scalar values 〈f(u),u〉〈f(u),u〉, u∈Su∈S. Not every probability measure on SS corresponds to a mapping in MM, and we conclude by examining several additional properties the representing measures must satisfy.