x1, x2, ... , xn are positive reals with sum less than 1. Show that nn+1x1x2 ... xn(1 - x1 - ... - xn) <= (x1 + x2 + ... + xn)(1 - x1)(1 - x2) ... (1 - xn).
Solution
Put xn+1 = 1 - (x1 + ... + xn). By AM/GM we have 1 - xi = x1 + ... + xi-1 + xi+1 + ... + xn >= n (x1...xn/xi)1/n. Multiplying the inequalities for i = 1, ... , n+1 we get (1 - x1) ... (1 - xn+1) ≥ nn+1 x1 ... xn, which is the required result.
© John Scholes
jscholes@kalva.demon.co.uk
30 Aug 2002