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from Part 1 - Bohr’s Problem and Complex Analysis on Polydiscs
Published online by Cambridge University Press: 19 July 2019
A holomorphic function f on the disc has a Taylor expansion with coefficients c_k. Bohr asked about the maximal 0<r<1 so that the supremum for |z|<r of ∑ | c_k z^k | is less than or equal to the supremum for |z|<1 of |f(z)|. Bohr’s power series theorem answers this question showing that r=1/3 is best possible. The n-th Bohr radius K_n is defined as the best r for which an analogous question holds for holomorphic functions on the n-dimensional polydisc. The sequence (K_n) is decreasing and tends to 0 as n goes to ∞ asymptotically like (\log n/n)^(1/2). The proof os this relies on an improved version of the polynomial Bohnenblust-Hille inequality (see Chapter 6), where the constant grows at most exponentially, and to get this a Khinchin-Steinhaus inequality for polynomials is needed, showing that all L_p norms of polynomials in n variables are equivalent.
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