The convergence of a sequence of polynomials with restricted zeros

Abstract

Suppose that 〈fn〉 is a sequence of polynomials, 〈fn(k)(0)〉 converges for every non-negative integer k, and that the limit is not 0 for some k. It is shown that if all the zeros of f1,f2,… lie in the closed upper half plane Imz≥0, or if f1,f2,… are real polynomials and the numbers of their non-real zeros are uniformly bounded, then the sequence converges uniformly on compact sets in the complex plane. The results imply a theorem of Benz and a conjecture of Pólya.