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if a, b, c>0, then prove: $\frac{a+b}{\sqrt{a^{2}+a b+b^{2}+b c}}+\frac{b+c}{\sqrt{b^{2}+b c+c^{2}+c a}}+\frac{c+a}{\sqrt{c^{2}+c a+a^{2}+a b}} \geq 2+\sqrt{\frac{a b+b c+c a}{a^{2}+b^{2}+c^{2}}}$

I have spent several hours in proving the inequality above with Hölder inequality. But all my efforts were in vain since it turned out that Hölder inequality cannot match the square root form on the right side of inequal symbol. I hope someone could help me solve it at least could offer me some hints. Thanks for your time and efforts in advance.

Though this inequality has been posed in this website 5 years ago, no complete answer using basic inequalities is presented. I am expecting some hints to prove this inequality using some basic inequalities rather than methods of analysis.

Aachen
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    Welcome to Math.SE. Can you please edit your post and add the description of your attempts at solving your question? It will be easier for somebody to help you out if we have some context and understand exactly what are you struggling with. – Ertxiem - reinstate Monica Apr 20 '19 at 12:32
  • This looks pretty cyclic to me. – Arthur Apr 20 '19 at 12:35
  • But my friend @Arthur, cyclic though it looks, it cannot be prove in the similar way as cyclic inequality is done. Take care! – Aachen Apr 20 '19 at 12:38
  • It was here: https://math.stackexchange.com/questions/673411 but without proof. – Michael Rozenberg Apr 20 '19 at 13:29
  • Though this inequality has been posed in this website 5 years, no complete answer using basic inequalities is presented. I am expecting some hints to prove this inequality using some basic inequalities rather than methods of analysis. – Aachen Apr 20 '19 at 13:38
  • It is true this is a duplicate, but the original question is really unanswered. What is there is no solution, its just hand waving. – Macavity Apr 20 '19 at 14:04
  • Perhaps, it is the right time to solve this problem by our combined efforts. So don't hesitate to pose any ideas about this inequality now! – Aachen Apr 20 '19 at 14:15
  • @Aachen I think, Holder does not help here. – Michael Rozenberg Apr 20 '19 at 14:22
  • @Michael Rozenberg So do you have any good ideas, my friend? – Aachen Apr 20 '19 at 15:47
  • I have, but I need to check it. – Michael Rozenberg Apr 20 '19 at 15:48
  • Thank you, my friend! May you offer some hints about your idea even if it is a half-baked one? So perhaps, the whole community may benefit from it and be inspired. – Aachen Apr 20 '19 at 15:55

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