Let x1, x2, ... , xn be positive reals with sum 1. Prove that x1/(2 - x1) + x2/(2 - x2) + ... + xn/(2 - xn) ≥ n/(2n - 1).
Solution
Assume x1 ≤ x2 ≤ ... ≤ xn. Then also 1/(2 - x1) ≤ 1/(2 - x2) ≤ ... ≤ 1/(2 - xn). Hence we may apply Chebyshev's inequality to get x1/(2 - x1) + x2/(2 - x2) + ... + xn/(2 - xn) ≥ (x1 + ... + xn)(1/(2 - x1) + ... + 1/(2 - xn) )/n. Now the harmonic mean inequality gives (1/(2 - x1) + ... + 1/(2 - xn) ) ≥ n2/( (2 - x1) + ... + (2 - xn) ) = n2/(2n - 1). Hence result.
© John Scholes
jscholes@kalva.demon.co.uk
11 Apr 2002