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\sum x_i^t \le \sum x_i ^{t+1} if \sum x_i \le \sum x_i ^2

Source: Polish second round 1995 p4

January 19, 2020
inequalitiesalgebran-variable inequalitySum

Problem Statement

Positive real numbers x1,x2,...,xnx_1,x_2,...,x_n satisfy the condition ∑i=1nxi≤∑i=1nxi2\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2 . Prove the inequality ∑i=1nxit≤∑i=1nxit+1\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1} for all real numbers t>1t > 1.
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