a1, a2, ... , an are real and b < (∑ ai)2/(n - 1) - ∑ ai2. Show that b < 2aiaj for all distinct i, j.
Solution
It is sufficient to show that (∑ ai)2/(n - 1) - ∑ ai2 ≤ 2a1a2. But this follows immediately from the Cauchy inequality for the two n-1 tuples:
a1 + a2, a3, a4, ... , an; and 1, 1, ... , 1:
(∑ ai)2 <= (n - 1)( (a1 + a2)2 + a32 + ... + an2) = (n - 1) ∑ ai2 + (n - 1) 2a1a2.
© John Scholes
jscholes@kalva.demon.co.uk
30 Nov 1999