# Prove $\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$ with $a,\,b,\,c> 0$

Let $$a,\,b,\,c$$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\dfrac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\dfrac{a}{\sqrt{b(\,c+ a\,)}}$$ I tried Holder and $$\lceil$$ https://artofproblemsolving.com/community/c6h194103p1065812 $$\rfloor$$ but it's too hard to access! I need to the hint, interest from MSE-ees and hope to see $$uvw$$ help here! Thanks a lot!

• At first glance, this looks to me like something for the rearrangement inequality. Have you tried applying that? – Arthur May 18 at 14:28
• @HaiDangel If it's true it's a very interesting problem. Where did you take this inequality? – Michael Rozenberg May 19 at 6:40
• @HaiDangel Where you took this problem? – Michael Rozenberg May 20 at 5:55