The sub-Bergman Hilbert spaces are analogues of de Branges-Rovnyak spaces in the Bergman space setting. We prove that the polynomials are dense in sub-Bergman Hilbert spaces. This answers the question posted by Zhu in the affirmative.2010 Mathematics Subject Classification. Primary 47B32.

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Sub-Bergman Hilbert spaces are analogues of de Branges-Rovnyak spaces in the Bergman space setting. They are reproducing kernel Hilbert spaces contractively contained in the Bergman space of the unit disk. K. Zhu analyzed sub-Bergman Hilbert spaces associated with finite Blaschke products, and proved that they are norm equivalent to the Hardy space. Later S. Sultanic found a different proof of Zhu's result, which works in weighted Bergman space settings as well. In this paper, we give a new approach to this problem and obtain a stronger result. Our method relies on the theory of reproducing kernel Hilbert spaces.

Abstract. We consider the Bohr radius Rn for the class of complex polynomials in one variable of degree at most n. It was conjectured by R. Fournier in 2008 that Rn = 1 3. We shall prove this conjecture is true in this paper.

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