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u^a v^b <= au + bv if u,v>=0 a,b>0 with a+b=1

Source: Spanish Mathematical Olympiad 1982 P6

December 23, 2022
algebrainequalities

Problem Statement

Prove that if u,vu, v are any nonnegative real numbers, and a,ba,b positive real numbers such that a+b=1a + b = 1, then uavb≤au+bv.u^a v^b \le au + bv.
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