Generalized Bohr radius for slice regular functions over quaternions
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Generalized Bohr radius for slice regular functions over quaternions Zhenghua Xu1 Received: 21 April 2020 / Accepted: 28 August 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we shall investigate some generalized Bohr radius in the non-commutative setting of quaternions. To be precise, two results of Enrico Bombieri are generalized by means of a new idea into the class of slice regular functions over quaternions. As an application, the Bohr radius is determined for p-symmetric slice regular functions over quaternions. Keywords Function of one hypercomplex variable · Bohr radius · Bombieri theorem · Quaternion Mathematics Subject Classification Primary 30G35 · Secondary 30B10 · 30H05
1 Introduction In the study of Dirichlet series, Harald Bohr discovered the following interesting phenomenon.
∑ n Theorem A (Bohr theorem) Let f (z) = +∞ n=0 an z be a holomorphic function in the open unit disk 𝔻 = {z ∈ ℂ∶ |z| < 1} such that |f (z)| ≤ 1 for all z ∈ 𝔻 . Then, the associated majorant function Mf is such that Mf (r) ∶=
+∞ ∑ n=0
|an |rn ≤ 1,
r = |z| ≤
1 . 3
Moreover, the constant 1/3, known as the Bohr radius, is the best possible.
This theorem was originally proved in 1914 by Bohr [11] for r ≤ 1∕6 and was refined thereafter by M. Riesz, Schur, and Wiener independently. In 1995, the Bohr theorem was applied by Dixon [19] to the long-standing problem of characterizing Banach
* Zhenghua Xu [email protected] 1
School of Mathematics, HeFei University of Technology, Hefei 230601, China
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Z. Xu
algebras satisfying the von Neumann inequality and then it obtained considerable attention by various researchers. In particular, the study of the Bohr radius for holomorphic functions in several complex variables and in Banach spaces becomes very active, see, e.g., [2, 7, 9, 10, 17, 27, 29]; in the non-commutative setting, see [30, 31] for the operator-valued generalizations of the Bohr theorem in the single variable case and [33–35] for free holomorphic functions in multivariable cases. For more detailed developments of this topic, the reader may refer to the recent survey [1] and references therein. In this paper, we shall focus on the generalized Bohr radius in the non-commutative algebra of quaternions. In 1955, Ricci [36] initiated the study of the Bohr radius for holomorphic functions f with fixed coefficient f(0), and in 1962, the Fields Medalist Bombieri [12] gave a partial answer in the case |f (0)| ≥ 1∕2 by establishing the following result.
Theorem ∑B (Bombieri theorem) Let a ∈ 𝔻 be such that |a| ∈ [1∕2, 1] . If n 𝔻 with |f (z)| ≤ 1 for all z ∈ 𝔻 , then f (z) = a + +∞ n=1 an z is a holomorphic function in |a| +
+∞ ∑ n=1
|an |rn ≤ 1,
r≤
1 . 1 + 2|a|
Moreover, the constant 1∕(1 + 2|a|) is the best possible. Note that the optimal radius in Theorem B for |a| < 1∕2 is still an open problem. Very recently, Bhowmik and Das [8] have obtained the following operator-valued analogue of the Bombie
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