An extremal problem for the geometric mean of polynomials

Let M0,, be the maximum of the geometric mean of all nth degree polynomials a,ekl which satisfy Jakl = 1, k=O, 1, n. We show the existence of certain polynomials Rn whose geometric mean is asymptotic to V/n, thus proving that M0,o is itself asymptotic to -\/n. Consider the class n of all nth degree polynomials Zk==0 akzk for which lakl=1, k=0, 1, , n. Following the usual notation, let r,~~~~~/ M((f) = (2rf)-l If (ei")I do) for r>0, and let Mo(f) = G(f) = exp ((27r)-1 log If(eto)I do), so that, in particular, for r=O, 1, and 2, Mr(f) is the geometric mean, arithmetic mean, and mean square of f, respectively. Now, forf c gn, and 0 V/n-c. Beller [1] noted that this Received by the editors May 22, 1972 and, in revised form, October 9, 1972. AMS (MOS) subject classfiJcations (1970). Primary 30A06, 42A04; Secondary 30A08.