An ¹ extremal problem for polynomials

it follows that M?(n+1)1"2. However, in order that (n+1)1/2 be (close to) the right answer the inequality M2 > (1/2r)f I EakZ | 2 dz| must be (close to) equality, which is to say, I Eakz41 must be close to constant. It was proved by Hardy that this minimum M, Mn, satisfies Mn<? c(n+ 1)1/2, c some absolute constant (see Zygmund [4]). Shapiro has even shown that the ak can be chosen real (i.e., equal to +1) and'the same estimate achieved (see Rudin [3]). Thus the order of magnitude of Mn is determined as -Vn. The deeper question regarding the limit of Mn/I/n remains unsettled. In terms of our original heuristic formulation it makes a vital difference whether Mn/Vn-1l or not. Some partial results in this direction have been obtained by Erdos and Littlewood [1]. Another extremal problem in the same spirit is the following: Among all polynomials for which