On the size of balls covered by analytic transformations

Two quantitative forms of the inverse function theorem giving estimates on the size of balls covered biholomorphically are proved for holomorphic mappings of a ball in a Banach space into the space. Also, a Bloch theorem for K-quasiconformal mappings on the open unit ball of a Banach space is given and some mapping properties of K-quasiconformal mappings are deduced.