Equality of norms for a class of Bloch and symmetrically weighted Lipschitz spaces of vector valued functions and derivation inequalities for Pick functions

Abstract

If X and Y are Banach spaces and f:BX→Y is Fréchet differentiable on the open unit ball BX of X, then for every operator monotone function φ:(−1,1)→R, which satisfies φ′′⩾0 on [a,b), [Formula presented] This generalizes Holland–Walsh–Pavlović criterium for the membership in Bloch type spaces for functions defined in the unit ball of a Banach space and taking values in another Banach space. We also established relations of the induced Bloch and Lipschitz spaces with other spaces of vector valued functions.