Cyclic product of positive real numbers each less than 1 is less than 1/4

Posted on January 27, 2015 by Sumant Sumant

Let x_1,x_2,...,x_n,n \ge 4 be non-negative real numbers that add up to 1 or \sum x_i =1 Prove that x_1x_2+x_2x_3+...+x_nx_1 \le \frac{1}{4}.

Well its easy to see that it is true for n \ge 4

Assume that they are all arranged about a circle, so that S_n = x_1x_2+x_2x_3+...+x_nx_1 can be regarded as a pair of adjacent numbers. Factoring the expression for S_4 and using the AM-GM inequality we get

x_1x_2+x_2x_3+x_3x_4+x_4x_1

x_2(x_1+x_3)+x_4(x_3+x_1)

(x_1+x_3)(x_2+x_4)

(x_1+x_3)(x_2+x_4)\le \frac{1}{4}(x_1+x_2+x_3+x_4)^2

(x_1+x_3)(x_2+x_4)\le \frac{1}{4} as \sum{x_i} = 1

Thus the statement is true for n=4

Assuming it is true for n = k-1

About Sumant Sumant

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