# Vector-valued holomorphic functions revisited

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**Abstract.** Let \(\Omega \subset \C\) be open,*X* a Banach space and \(W\subset X^\prime\). We show that every \(\sigma (X,W)\mbox{-holomorphic function } f: \Omega \to X\) is holomorphic if and only if every \(\sigma(X,W)\mbox{-bounded}\) set in*X* is bounded. Things are different if we assume *f* to be locally bounded. Then we show that it suffices that \(\varphi \circ f\) is holomorphic for all \(\varphi \in W\), where *W* is a separating subspace of \(X^\prime\) to deduce that *f* is holomorphic. Boundary Tauberian convergence and membership theorems are proved. Namely, if boundary values (in 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).

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