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

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 $$1 \leq s \leq \infty$$.

### 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
Full Text:

### References:

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