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sum \frac{ab}{\sqrt[4]{3c^2+16}} \le 4/3 \sqrt[4]{12} if a+b+c=4, a,b,c>0

Source: Mediterranean Mathematical Olympiad 2020 P3 MMC

September 23, 2020

algebrainequalitiesMediterranean

Problem Statement

Prove that all postive real numbers

a,b,c

with

a+b+c=4

satisfy the inequality

\frac{ab}{\sqrt[4]{3c^2+16}}+ \frac{bc}{\sqrt[4]{3a^2+16}}+ \frac{ca}{\sqrt[4]{3b^2+16}} \le\frac43 \sqrt[4]{12}