Published online by Cambridge University Press: 20 November 2018
The following result has been conjectured by Dr. Birch. Let z1 z2, . . . , zn be any n complex numbers such that
(1)
Then
(2)
attains its greatest value when the z are at the vertices of a regular n-sided polygon inscribed in the circle |z| =1.
It seems to be difficult to prove this but Dr. Birch informs me that some work by Mullholland shows that the result is false for large n. I can, however, prove that the result is true for n = 3, and then Δ ≤ 27. The suggested general result would be Δ ≤ nn.
1 ”Inequalities between the geometric mean difference and the polar moments of a plane distribution,” Journal of the London Mathematical Society, 33 (1958) 260-269.