On Bohr radii of finite dimensional complex Banach spaces. (English) Zbl 1435.46034

Defant, Andreas; Mastyło, Mieczysław; Schlüters, Sunke

Funct. Approximatio, Comment. Math. 59, No. 2, 251-268 (2018).

If \(f: \mathbb{D} \to \mathbb{C}\) is holomorphic, let us denote \(f=\sum_{n=0}^{\infty} c_{n} (f) z^{n}\). Bohr’s power series theorem shows that \begin{align*} \sup \Big\{ 0 < r< 1 \colon \sup_{\vert z \vert < r} & \sum_{n=0}^{\infty} \vert c_{n} (f) z^{n} \vert \leq \sup_{\vert z \vert < 1} \Big\vert \sum_{n=0}^{\infty} c_{n} (f) z^{n} \Big\vert \\ & \text{ for every bounded holomorphic } f : \mathbb{D} \to \mathbb{C}\Big\} = \frac{1}{3} \,. \end{align*} Aiming at extending this result to several variables motivated H. P. Boas and D. Khavinson to define in [Proc. Am. Math. Soc. 125, No. 10, 2975–2979 (1997; Zbl 0888.32001)] the multidimensional Bohr radius. Later, the Bohr radius of an arbitrary \(n\)-dimensional Banach space \(X=(\mathbb{C}^{n}, \Vert \cdot \Vert_{X})\) was defined in [A. Defant et al., J. Reine Angew. Math. 557, 173–197 (2003; Zbl 1031.46014)] in an analogous way as \begin{align*} K(B_{X})=\sup \Big\{ 0 < r< 1 \colon \sup_{\Vert z \Vert_{X} < r} &\sum_{\alpha \in \mathbb{N}_{0}^{n}} \vert c_{\alpha} (f) z^{\alpha} \vert \leq \sup_{\Vert z \Vert_{X} < 1} \Big\vert \sum_{\alpha \in \mathbb{N}_{0}^{n}}c_{\alpha} (f) z^{\alpha} \Big\vert \\ & \text{ for every bounded holomorphic } f : B_{X} \to \mathbb{C} \Big\} \,. \end{align*}
If \(X\) is a symmetric Banach sequence space, its \(n\)-th section \(X^{n}\) is defined as the span of \(\{e_{1}, \ldots , e_{n} \}\), the first \(n\) canonical vectors. In this paper, the authors give estimates for \(K(B_{X^{n}})\) depending on the concavity/convexity constants of \(X_{n}\). More precisely, the two main results show that, for \(1 \leq r \leq 2\), we have \[ \frac{1}{M_{(r)}(X^{n})} \, \frac{(\log n)^{1-\frac{1}{r}}}{n^{1-\frac{1}{r}}} \prec K(B_{X^{n}}) \prec M_{(r)}(X^{n}) \frac{(\log n)^{1-\frac{1}{r}}}{n \varphi_{X}(n)} \] (where \(M_{(r)}(X^{n})\) is the \(r\)-concavity constant of \(X^{n}\) and \(\varphi_{X}\) is the fundamental function of \(X\)) and, if \(X\) is \(2\)-convex, then \[ 1 \leq \liminf_{n} \frac{K(B_{X^{n}})}{\sqrt{\frac{\log n}{n}}} \leq \limsup_{n} \frac{K(B_{X^{n}})}{\sqrt{\frac{\log n}{n}}} \leq M^{(2)} (X) \] (where \(M^{(2)}(X^{n})\) is the \(2\)-convexity constant of \(X\)).
As an application, rather accurate estimates are given for \(K(B_{\ell_{r,s}^{n}})\), the Bohr radius of the \(n\)-dimensional Lorentz and Marcinkiewicz spaces \(\ell_{r,s}^{n}\) with \(1 < r < \infty\) and \(1 \leq s \leq \infty\).

Reviewer: Pablo Sevilla Peris (Valencia)

MSC:

46G25	(Spaces of) multilinear mappings, polynomials
46B07	Local theory of Banach spaces
52A21	Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
32A05	Power series, series of functions of several complex variables

Keywords:

Bohr radius; power series; polynomials; Banach sequence spaces

Citations:

Zbl 0888.32001; Zbl 1031.46014

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