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An inequality problem from China TST

Source: 2017 China TST 4 Problem 5

March 22, 2017
inequalitiesTST

Problem Statement

Given integer m≥2m\geq2,x1,...,xmx_1,...,x_m are non-negative real numbers,prove that:(m−1)m−1(x1m+...+xmm)≥(x1+...+xm)m−mmx1...xm(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_mand please find out when the equality holds.
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