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These are two inequalities from my assignments. I don't know if it is too difficult or I am not so good at inequalities but please help me with full answers.

  1. Let $a$,$b$,$c$ be three real positive numbers, prove that

enter image description here

So first, to avoid typing too much character, I may write the LHS of the first problem in the form of $\sum\frac{a(a−2b+c)}{a×b+1}$, which I transformed to $\sum\frac{a^2+1}{a×b+1} + \sum\frac{a×c+1}{a×b+1}−6.$

It is easy to prove using a.m-g.m inequality that $\sum\frac{a×c+1}{a×b+1}\geq3.$

It is also clear that $\sum\frac{a^2+1}{a×b+1}\geq3$ by Cauchy-Schwartz Inequality. Q.E.D –

  1. Let $x$,$y$,$z$ be three non-negative real numbers, prove that: enter image description here

The second problem is a little more complicated, I think so I haven't finished it yet, but still, this is what I have worked on:

$\sum\frac{x}{\sqrt{x^2+3}}\leq\frac{3\sqrt3}{8}\sqrt{x^2+y^2+z^2}+\frac{3}{8}.$

According to the Cauchy-Schwartz inequality, $\sqrt{(x^2+3)(1+3)}\geq x+3$,

$\sqrt{3(x^2+y^2+z^2)}\geq x+y+z$; further transformations yield,

$\frac{3}{8}\sum(x+3)+\sum\frac{6}{x+3}\geq9$. This seems so obvious but how do I finish it?


POST SCRIPT I really look forward to you posting your full answers rather than comment on this post. Please help with this

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    $\begingroup$ Post these inequalities in different topics and show please your attempts. $\endgroup$ – Michael Rozenberg Mar 9 '20 at 5:00
  • $\begingroup$ where should i post these inequalities? $\endgroup$ – Nikola Tolzsek Mar 9 '20 at 5:09
  • $\begingroup$ Just open another topic. By the way, I solved your first problem. If you want to see my solution, show please your attempts. $\endgroup$ – Michael Rozenberg Mar 9 '20 at 5:12
  • $\begingroup$ I solved also your second problem. If you want to see my solution, show please your attempts. Otherwise, this topic would be deleted. $\endgroup$ – Michael Rozenberg Mar 9 '20 at 23:30
  • $\begingroup$ okay, I will write it below. $\endgroup$ – Nikola Tolzsek Mar 10 '20 at 2:02
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The first problem.

Your proof is beautiful!

My solution:

We need to prove that: $$\sum_{cyc}\frac{a^2+ac-2ab}{ab+1}\geq0$$ or $$\sum_{cyc}\frac{a^2+ac+2}{ab+1}\geq6.$$ Now, by C-S $$\sum_{cyc}\frac{a^2+ac+2}{ab+1}\geq\frac{\left(\sum\limits_{cyc}(a^2+ac+2)\right)^2}{\sum\limits_{cyc}(a^2+ac+2)(ab+1)}.$$ Thus, it's enough to prove that: $$\frac{\left(\sum\limits_{cyc}(a^2+ac+2)\right)^2}{\sum\limits_{cyc}(a^2+ac+2)(ab+1)}\geq6$$ or $$\sum_{cyc}(a^4-4a^3b+2a^3c+3a^2b^2-2a^2bc)+6\sum_{cyc}(a^2-ab)\geq0$$ and since $$6\sum_{cyc}(a^2-ab)=3\sum_{cyc}(a-b)^2\geq0,$$ it's enough to prove that $$\sum_{cyc}(a^4-4a^3b+2a^3c+3a^2b^2-2a^2bc)\geq0,$$ which is smooth.

The second problem.

By your work it's enough to prove that: $$\frac{3}{8}\sum_{cyc}(x+3)+\sum_{cyc}\frac{6}{x+3}\geq9,$$ which is true by C-S and AM-GM: $$\frac{3}{8}\sum_{cyc}(x+3)+\sum_{cyc}\frac{6}{x+3}=\frac{3}{8}\left(\sum_{cyc}(x+3)+16\sum_{cyc}\frac{1}{x+3}\right)\geq$$ $$\geq \frac{3}{8}\left(\sum_{cyc}(x+3)+\frac{144}{\sum\limits_{cyc}(x+3)}\right)\geq\frac{3}{8}\cdot2\cdot\sqrt{144}=9.$$

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