Ineq!⠀⠀⠀
by cjquines0, Dec 29, 2016, 9:49 AM
MOP 2002 wrote:
Let 
be positive real numbers. Prove that 
be positive real numbers. Prove that 
.
follows from 
and 
Applying Karamata’s on 
gives the desired inequality.3 Comments
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by
agbdmrbirdyface, Dec 29, 2016, 10:45 PM
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by
zephyrcrush78, Dec 30, 2016, 1:32 AM
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Assume WLOG 
since the inequality is homogenous. We have to prove ![$$\sum_{cyc}a\cdot\sqrt[3]{\frac{1}{a(1-a)^2}}\ge\frac{3}{\sqrt[3]{4}}$$](//latex.artofproblemsolving.com/d/0/6/d063d0e1d41761e6c558c34c1e7c2e7ff73d27b2.png)
Consider function ![$$\forall_{x\in(0;1)}\ f(x)=\sqrt[3]{\frac{1}{x(1-x)^2}}$$](//latex.artofproblemsolving.com/4/b/5/4b5706f2a62f7ff06010094cf0a32a2dbb025f9b.png)
Then 
and
We have proven that 
is a convex function. Thus
where 
Of course 
so 
exists in the given domain. We are left with proving 
That easily follows from AM-GM:
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since the inequality is homogenous. We have to prove ![$$\sum_{cyc}a\cdot\sqrt[3]{\frac{1}{a(1-a)^2}}\ge\frac{3}{\sqrt[3]{4}}$$](http://latex.artofproblemsolving.com/d/0/6/d063d0e1d41761e6c558c34c1e7c2e7ff73d27b2.png)
Consider function ![$$\forall_{x\in(0;1)}\ f(x)=\sqrt[3]{\frac{1}{x(1-x)^2}}$$](http://latex.artofproblemsolving.com/4/b/5/4b5706f2a62f7ff06010094cf0a32a2dbb025f9b.png)
Then 
and
We have proven that 
is a convex function. Thus
where 
Of course 
so 
exists in the given domain. We are left with proving 
That easily follows from AM-GM:
Awaiting for likes.To share with readers my favorite problem I came across today :) (Shout for contrib.)
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