Maximal values of symmetric functions in distances between points

Abstract

In this note we find the maximal values of several symmetric functions in the variables which are the squares of distances |zi − z j| , 1 i < j d , between some d complex points z1, . . . ,zd in the unit disc. We compute the maximums of σm , for m = 1,2,3,4 , explicitly and find the conditions on z1, . . . ,zd under which those maximal values are attained. This problem is motivated by an inequality of Cassels (1966) and a subsequent conjecture of Alexander. Mathematics subject classification (2010): 52A40, 11R06.