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

Marc Technau

Given a polynomial ${\sum }_{\hat{I}}½a_{\hat{I}}½X^{\hat{I}}½$ of degree $<d$, bounded by one on the unit disk, how large can $|a_0+a_1+\dots +a_n|$ ($n<d$) get? This question dates back at leas…