Shapiro's problem on polynomials with large partial sums of coefficients

Abstract

Given a polynomial $\sum_\nu a_\nu X^\nu$ of degree $<d$, bounded by one on the unit disk, how large can $\lvert a_0+a_1+\ldots+a_n \rvert$ ($n<d$) get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D. J. Newman gave an exact answer for $d=2(n+1)$, but there does not seem to have been further progress on the question since. We study variations on this theme, obtaining exact answers for some related coefficient sums, and answer the original question in an asymptotic sense, provided that $n$ is not too large in terms of $d$. The latter is achieved via a quantitative Enestr\"om--Kakeya theorem, while the former is based on certain identities for carefully selected Lagrange interpolators. From the interpolation approach we also obtain a general inequality for coefficient sums $\lvert t_0 a_0 + \ldots + t_{d-1} a_{d-1} \rvert$ for arbitrary complex numbers $t_0,\ldots,t_{d-1}$. This inequality fails to be sharp in general, yet it is in some cases and also yields non-trivial bounds for Shapiro's problem for some choices of $n$ and $d$.