Conditions for generating a nonvanishing bounded analytic function
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- by J. H. Mantel PDF
- Proc. Amer. Math. Soc. 66 (1977), 62-64 Request permission
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 $\log [\Sigma _{i = 1}^n|{f_i}|]$ 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 ${H^\infty }$.References
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 66 (1977), 62-64
- MSC: Primary 30A76
- DOI: https://doi.org/10.1090/S0002-9939-1977-0457732-6
- MathSciNet review: 0457732