The minimum modulus of polynomials

In answer to a problem of Erdos and Littlewood we produce an nth degree polynomial, P(z), with coefficients bounded by 1 satisfying |P(z)I>CC/n for all z on |z = 1 (C is a positive absolute constant). Littlewood and Erdos, independently, asked whether polynomials, P(z), of degree n could exist having all coefficients bounded by 1 and satisfying min,z= 1 |P(z)| > Cxn (C a fixed positive constant). Clunie [2] gave a very ingenious construction of a polynomial purporting to do this job, but it was based on a result of Littlewood's [3] which later proved erroneous. Littlewood claimed that, as r 1, min ninznj = Q(i /2 zl=r n=l but his reasoning had a flaw which was discovered by Erdos and Carroll. Indeed a careful examination of his method shows the very opposite, that min | ninzn|= o(i-r)-1/2 lzl=-r n= I In this note we give an extremely simple construction of a polynomial which does have the desired properties. Consider the function /(O) defined as exp(in 302) in [ur, u1 and extended to have period 2,r. (3 is a small but fixed positive number.) Write K = [n/2], let t(O) be the Kth Cesaro partial sum of the Fourier series of /(O), io K and finally set P(e ) = v/n3eZKt(0). Clearly P(z) is a polynomial of degree /n (I 40 6log 6 1) for all z on lz I = I . Received by the editors December 10, 1973. AMS (MOS) subject classifications (1970). Primary 26A75, 26A82, 30A06.