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An abstract approach to Bohr’s phenomenon. (English) Zbl 0958.46015
In 1914 H. Bohr discovered that there exists \(r\in (0,1)\), namely \(r=1/3\), such that for any holomorphic function \(f(z)=\sum_{k=0}^ \infty c_kz^k\) on the unit disk \(\{ z\in {\mathbb C}:|z|<1\}\), and having its modulus there less than \(1\), the inequality \(\sum_{k=0}^ \infty |c_kz^k|<1\) is valid in the disk \(\{ z\in {\mathbb C}:|z|<r\}\), and this constant \(r\) can not be improved [see H. Bohr, “A theorem concerning power series”, Proc. Lond. Math. Soc., II. Ser. 13, 1-5 (1914)]. The same constant \(r\), such that \(\sum_{k=0}^ \infty |c_kz^k|<2f(0)\) is valid in the disk \(\{ z\in {\mathbb C}:|z|<r\}\), exists for all holomorphic functions \(f(z)=\sum_{k=0}^ \infty c_kz^k\) on the unit disk, with positive real part and \(f(0)>0\), and this constant (i.e. \(r=1/3\)) cannot be improved. Multidimensional analogues of Bohr’s result for Taylor expansions of functions on complete Reinhardt domains were considered in a preprint paper of the first author, and in [H. P. Boas and D. Khavinson, Proc. Am. Math. Soc. 125, No. 10, 2975-2979 (1997; Zbl 0888.32001)].
In the paper under review, the generalizations of these results are obtained in a more general setting and in the more abstract spirit of functional analysis. Let \(H(M)\) be the space of holomorphic functions on a complex manifold \(M\), and \(\|\cdot\|_r,\;r\in (0,1)\), be a one-parameter family of seminorms in \(H(M)\), that are continuous with respect to the topology of uniform convergence on compact subsets of \(M\), and \(\|\cdot \|_{r_1}\leq\|\cdot\|_{r_2}\) if \(r_1\leq r_2\). The authors formulate the following problems.
Problem \(B_1\). Do there exist an \(r\in (0,1)\) and a compact \(K\subset\subset M\) such that \(\|f\|_r\leq\sup_K |f(z)|\) \(\forall f\in H(M)\)?
Problem \(B_2\). Find \(\sup\{ r:\|f\|_r\leq\sup_M |f(z)|\) \(\forall f\in H(M)\) and bounded}. The authors call the finite solution of Problem \(B_2\) the Bohr radius. In fact, the original Bohr’s result says that if \(M\) is the unit disk and \(\|f\|_r=\sup_{|z|\leq r}\sum_n |c_nz^n|\) where \(f(z)=\sum_nc_nz^n\) is the Taylor expansion of \(f\), then Problem \(B_2\) has solution \(r=1/3\), i.e. in this case the Bohr radius equals \(1/3\).
Problem \(PB_1\). Given \(z_0\in M\), is there an \(r\in (0,1)\) such that for \(\|f\|_r\leq 2f(z_0)\) \(\forall f\in P=\{ f\in H(M): \operatorname {Re} f(z)>0, f(z_0)>0\}\)?
Problem \(PB_2\). In case Problem \(PB_1\) has a positive solution find \(\sup\{ r:\|f\|_r\leq 2f(z_0)\) \(\forall f\in P\}\).
In this case the solution of Problem \(PB_2\) is also called the Bohr radius. The authors give certain conditions on families of seminorms \(\|\cdot\|_r\) in \(H(M)\) that are sufficient for the solvability of Problem \(B_1\) (resp., Problem \(PB_1\)). They point out some cases where estimates for Bohr radii of Problems \(B_2\) and \(PB_2\) are obtained. They also show that under some very general assumptions on seminorms (as for the original problems in the unit disk case) the Bohr radii of Problems \(B_2\) and \(PB_2\) coincide. However, in general, they do not coincide. Even for the disk case, the example is given when Bohr radii of Problems \(B_2\) and \(PB_2\) are different. Also, the examples are constructed when both Problems \(B_1\) and \(PB_1\) have negative solution; when Problem \(B_1\) has negative solution, but Problem \(PB_1\) has positive solution; when Problem \(B_1\) has positive solution, but Problem \(PB_1\) has negative solution.

MSC:
46E10 Topological linear spaces of continuous, differentiable or analytic functions
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
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