I hope someone can help me giving a hint or sth for my inequality, which I'm trying to solve now for some days. I want to show that $$\frac{2}{\sqrt{\vphantom{\large A}1+c}}\ \leq\ \frac{1}{\sqrt{1+c\,\left(\frac{c\ +\ \sqrt{\vphantom{\Large A}c^{2}\ -\ 4\,}\,}{\vphantom{\Large A}2}\right)^{3}}} + \frac{1}{\sqrt{1+c\,\left(\frac{c\ -\ \sqrt{\vphantom{\Large A}c^{2}\ -\ 4\,}\,}{\vphantom{\Large A}2}\right)^{3}}}\,,\quad \forall\ c \geq 2$$
From plotting its easy to see and Mathematica/Maple also gave me the solution c>2, but thats not a real proof. Still from this, I think there must be a way to show it.
Besides just trying to show the inequality i tried showing that the difference of these terms is an injective function (for c=2 there is equality, which is a problem for many approximations). However the resulting terms dont really get easier to handle...
Other than that I tried to use some meanvalue inequalities of harmonic mean, geometric mean, quadratic mean. The problem is that the inequalitiy between HM and GM is already too rough.
So at the moment im quite stuck and feel out of ideas how to approach the problem. It would be really nice if someone has some ideas (i hope i dont need a full solution :) )
P.S.: please excuse my bad english, I'm no native speaker.
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