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HELP Inequality problem (from Studying Math FB page)

Let \(a, b, c \) be non-negative real numbers. Prove that

\[ \frac{ ab}{a+4b+4c} + \frac{bc}{b+4c+4a} + \frac{ ac}{c+4a+4b} \leq \frac{ a+b+c}{9} \]

Hi! :) I saw this difficult inequality problem on the Studying Math FB Page but haven't been able to solve it for a very long time! (No copyright intended!) Any help would be appreciated. Thanks! :)

Note by Happy Melodies
3 years, 5 months ago

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I have some solutions :) Dinesh Chavan · 3 years, 1 month ago

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@Dinesh Chavan Could you please give it out?I tried homogenizing, linearization.The only other ways are expanding and then using some inequality like Muirhead or Holder. Bogdan Simeonov · 3 years, 1 month ago

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Maybe since the inequality it homogeneous creating a condition like \(a+b+c\) might help? My other thought is try using Jensen's inequality because it's normally really useful with cyclic inequalities like this, and the \(9\) looks Jensen-esque. You could always just multiply everything up and AM-GM/ Murihead's? Daniel Remo · 3 years, 3 months ago

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@Daniel Remo Sure! I will try it out :) Thanks! Happy Melodies · 3 years, 3 months ago

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