Remarks on the Bohr phenomenon
In 1914 Harald Bohr published the following surprising result: If \(f(z)=\sum a_n z^n\) is an analytic functions on the unit disc such that \(| f(z)| \leq 1\) for each \(z\) in the disc, then \(\sum | a_n z^n | \leq 1\) when \(| z| \leq 1/3\), and moreover the radius \(1/3\) is best possible. The authors note that no such Bohr phenomenon occurs if the space \(H^{\infty}\) is replaced by a Hardy space \(H^q, 0 < q < \infty\). In fact, there is no \(0<r<1\) such that \(\sum | a_n| r^n \leq 1\) for each \(f(z)=\sum a_n z^n\) in the closed unit ball of \(H^q\). \textit{P. B. Djakov} and \textit{M. S. Ramanujan}, [J.\ Anal.\ 8, 65--77 (2000; Zbl 0969.30001)], studied the growth of the series \(\sum | a_n| ^p r^n\) for bounded analytic functions on the disc. In Section 2 of the present article, the authors study the growth of \(\sum | a_n| ^p r^n\) as \(r\) tends to \(1\) for analytic functions in the unit ball of \(H^s, s>0\), particularly when \(0<p<2\). They observe that the case \(p \geq 2\) is open except for those trivial cases covered by the Hausdorff-Young theorem. In Section 3 they analyze the Bohr phenomenon for general Banach spaces \(X\) of analytic functions such that the polynomials are dense in \(X\), the point evaluation at each point of the disc is continuous and every non-constant function \(f\) in the closed unit ball of \(X\) satisfies that \(| f(0)| < 1\). A nice characterization is obtained and several examples are presented. Finally, the results are extended to several variables and, as a consequence, general Schwarz-Pick type estimates for bounded analytic functions are presented. Recently several authors have studied Bohr's theorem in higher dimension: given a finite dimensional Banach space \(X=(C^k,| | .| | )\), find the largest radius \(K(B_X)\), called the Bohr radius of the unit ball of \(X\), such that if \(| \sum a_{\alpha} z^{\alpha} | \leq 1\) for \(| | z| | <1\), then \(\sum | a_{\alpha} z^{\alpha}| \leq 1\) when \(| | z| | <K(B_X) \). A result of \textit{S. Dineen} and \textit{R. M. Timoney}, [Stud. Math. 94, No.3, 227-234 (1989; Zbl 0713.46005)], and \textit{H. P. Boas} and \textit{D. Khavinson}, [Proc. Am. Math. Soc. 125, No. 10, 2975--2979 (1997; Zbl 0888.32001)], states that the scalar sequence \((K(B_{l^k_{\infty}}))_k\) tends to 0 as \(k\) tends to infinity, and that the decay is essentially like \((1/k^{1/2})_k\). In a series of interesting papers, \textit{A. Defant, D. García} and \textit{M. Maestre} [J. Reine Angew. Math. 557, 173--197 (2003; Zbl 1031.46014) and J. Approximation Theory 128, No. 1, 53--68 (2004; Zbl 1049.32001)], have linked this circle of ideas to the local Banach space theory, in particular with asymptotic estimates of unconditional basis constants for spaces of homogeneous polynomials.
scientific article; zbMATH DE number 2104110
scientific article