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Prove this inequality on trigonometric function tan inverse

Source: 2019 Jozsef Wildt International Math Competition-W. 14

May 18, 2020
trigonometryinequalitiesfunction

Problem Statement

If aa, bb, c>0c > 0; ab+bc+ca=3ab + bc + ca = 3 then: 4(tan⁡−12)(tan⁡−1(abc3))≤πtan⁡−1(1+abc3)4\left(\tan^{-1} 2\right)\left(\tan^{-1}\left(\sqrt[3]{abc}\right)\right) \leq \pi \tan^{-1}\left(1 + \sqrt[3]{abc}\right)
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