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),(1)supa,b∈BX,a≠b⁡‖f(a)−f(b)‖φ′(‖a‖)‖a−b‖φ′(‖b‖)=supa∈BX⁡‖Df(a)‖φ′(‖a‖). 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.