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On a Discriminant Inequality

Published online by Cambridge University Press:  20 November 2018

L. J. Mordell
L. J. Mordell*
Affiliation:
Mount Allison University, Sackville, New Brunswick, Canada St John s College, Cambridge, England
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 699 - 704
Copyright
Copyright © Canadian Mathematical Society 1960

References

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.