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Σ|x_n-y_n| \leq 2- min {x_i/y_i} - min {y_i/x_i}, if Σx_i=Σy_i=1

Source: JBMO Shortlist 2014 A9

April 24, 2019

algebraminimuminequalitiespositive real

Problem Statement

Let

n

a positive integer and let

x_1, \ldots, x_n, y_1, \ldots, y_n

real positive numbers such that

x_1+\ldots+x_n=y_1+\ldots+y_n=1

. Prove that:

|x_1-y_1|+\ldots+|x_n-y_n|\leq 2-\underset{1\leq i\leq n}{min} \;\dfrac{x_i}{y_i}-\underset{1\leq i\leq n}{min} \;\dfrac{y_i}{x_i}