I am just wondering, is it true that

$$6 \sqrt{-3 t (1+t)+\sqrt{4+12 t\ \ }\ \ }$$

$$\le\left(\frac{3 t (1+t)+\sqrt{4+12 t}}{1+t+t^2}\ \ \right)^{3/2}+4\sqrt{2}(1-t)^{3/2}$$

for all $\displaystyle t\in[0,1]$?

Thanks!

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Oh, just what I was wondering too! :) – Hans Lundmark Oct 19 '11 at 6:27
(What I mean is that it might be helpful to provide some context. Is there any reason to believe it to be true?) – Hans Lundmark Oct 19 '11 at 6:27
Since both sides are positive, you can square each side and eliminate some of the roots... and then if you keep squaring it the right way, you can reduce the problem to a polynomial inequality...Which may or may not be easy, but will probably be easier than this. – N. S. Oct 19 '11 at 16:54

I did not mess around with the function yet. Do you need a formal proof or what is your purpose for this inequality? A simple Mathematica plot shows that the inequality is most likely true:

Also Mathematica can show your inequality is true:

Reduce[a[t] <= b[t] && 0 <= t <= 1,t]


returns

0 <= t <= 1

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Just curious - in what context did this problem arise? This is a difficult inequality to ponder just for the heck of it. If the result has some motiviation, it is easier to get enthusiastic. – Chris Leary Oct 19 '11 at 17:09