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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Conditions for generating a nonvanishing bounded analytic function


Author: J. H. Mantel
Journal: Proc. Amer. Math. Soc. 66 (1977), 62-64
MSC: Primary 30A76
MathSciNet review: 0457732
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Abstract: B. A. Taylor and L. A. Rubel have posed the problem of finding necessary and sufficient conditions on a set of given functions $ {f_1},{f_2}, \ldots ,{f_n}$ in $ {H^\infty }$ such that there exist functions $ {g_1},{g_2}, \ldots ,{g_n}$ in $ {H^\infty }$ with $ \Sigma _{i = 1}^n{f_i}{g_i} \ne 0$ in the open unit disc. L. A. Rubel has conjectured that a necessary and sufficient condition is that there exist a harmonic minorant of $ \log [\Sigma _{i = 1}^n\vert{f_i}\vert]$ in the open unit disc. The major result of this paper proves that the conjecture is true if one of the given functions $ {f_1},{f_2}, \ldots ,{f_n}$ has a zero set which is an interpolation set for $ {H^\infty }$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0457732-6
PII: S 0002-9939(1977)0457732-6
Keywords: Blaschke product, $ {H^\infty }$, interpolation sequence for $ {H^\infty }$, harmonic minorant, zero set, generating a nonvanishing bounded analytic function
Article copyright: © Copyright 1977 American Mathematical Society