Conditions for generating a nonvanishing bounded analytic function

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 ${H^\infty }$.