Abstract.In a recent paper the first author gave an explicit construction of a sequence of polynomials having their zeros on the unit circumference which converge boundedly to a given bounded zero-free analytic function in the unit disk. In this paper we find the best possible uniform bound for such approximating polynomials and construct a sequence for which this bound is attained. The method is also applied to approximation of an analytic function in the unit disk by rational functions whose poles lie on the unit circumference. Some open problems are discussed.