The convergence of a sequence of polynomials and the distribution of their zeros

Abstract

Suppose that $⟨f_n⟩$ is a sequence of polynomials, $⟨f_n^{(k)}(0)⟩$ 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 $f_1,f_2,\dots$ lie in the closed upper half plane $\mathrm{Im}\mathrm{z}\ge 0$, or if $f_1,f_2,\dots$ 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.