Abstract
Bohr radius and its asymptotic value for holomorphic functions in higher dimensions
Highlights
Introduction and the main resultsLet X be a complex Banach space and G ⊂ X, Ω ⊂ C be two domains
Motivated by [15, Corollary 3.2], we show in the last theorem of this article that [17, Theorem 2.1] extends for the holomorphic functions defined on a bounded balanced domain G in any complex Banach space X
Suppose X is a complex Banach space, G ⊂ X is a bounded balanced domain and f : G → D is a holomorphic function with an expansion (1) in a neighborhood of x0 = 0
Summary
Let X be a complex Banach space and G ⊂ X , Ω ⊂ C be two domains. For any holomorphic mapping f : G → Ω, let Dk f (x) denote the kth Fréchet derivative (k ∈ N) of f at x ∈ G, which is a bounded symmetric k-linear mapping from k i =1. Motivated by [15, Corollary 3.2], we show in the last theorem of this article that [17, Theorem 2.1] extends for the holomorphic functions defined on a bounded balanced domain G in any complex Banach space X. Suppose X is a complex Banach space, G ⊂ X is a bounded balanced domain and f : G → D is a holomorphic function with an expansion (1) in a neighborhood of x0 = 0. [17, the Theorems 2.3, 2.6, 2.7 and the Corollary 2.8] can be proved in sharp form for the holomorphic functions defined on a bounded balanced domain of a complex Banach space X in a similar manner as in Theorem 4 It may be noted that the other Bohr-like inequalities, i.e. [17, the Theorems 2.3, 2.6, 2.7 and the Corollary 2.8] can be proved in sharp form for the holomorphic functions defined on a bounded balanced domain of a complex Banach space X in a similar manner as in Theorem 4
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