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A bound for functions defined on a normed space

Let X be a complex normed space and f a complex valued function defined on X. Assume that is an entire function for all A, B ∊ X and there is a scalar monotone non-decreasing function G defined on [0, ∞), such that |f(A)| ⩽ G(‖A‖X)(A ∊ X). It is proved that Applications of this inequality to perturbations of the regularized determinants of the von Neumann–Schatten operators are also discussed.