Vector-valued holomorphic functions revisited

Let Omega subset of C be open, X a Banach space and W subset of X'. We show that every sigma(X, W)-holomorphic function f : Omega --> X is holomorphic if and only if every sigma(X, W)-bounded set in X is bounded. Things are different if we assume f to be locally bounded. Then we show that it suffices that phi circle f is holomorphic for all phi is an element of W, where W is a separating subspace of X' to deduce that f is holomorphic. Boundary Tauberian convergence and membership theorems are proved. Namely, if boundary values tin a weak sense) of a sequence of holomorphic functions converge/belong to a closed subspace on a subset of the boundary having positive Lebesgue measure, then the same is true for the interior points of Omega, uniformly on compact subsets. Some extra global majorants are requested. These results depend on a distance Jensen inequality. Several examples are provided (bounded and compact operators; Toeplitz and Hankel operators; Fourier multipliers and small multipliers).