Conditional inequality BM

[ins-]

Let $a$, $b$, $c$ are positive real numbers and $abc=1$. Prove that
$\frac{a^2}{b^2+c}+\frac{b^2}{c^2+a} + \frac{c^2}{a^2+b} \geq \frac{a}{b+c^2} + \frac{b}{c+a^2}+\frac{c}{a+b^2}$.

(Оффтоп)

What I would like to know is:
1. Is this problem correct?
2. How can be proved?
3. What is its level of difficulty?
4. Is it well known?

[arqady]

Ну шлёпнуть можно что угодно. А другие пусть доказывают или опровергают.

[ins-]

You are correct I cannot prove this. A friend of mine said a bulgarian mathematician solved my previous inequality in 3 ways. They said me it was easy and I tried to find a beautiful, easy looking but harder inequality. I played with Excel long time putting different values and I think it is true (at least it was impossible for me to find counterexample). About the solution what I'm thinking it probably can be solved with some case analysis. If it is possible there should be more beautiful solution but it is probably not easy to be found.

[arqady]

ins- в сообщении #555804 писал(а):

A friend of mine said a bulgarian mathematician solved my previous inequality in 3 ways.

Тот факт, что Ваше неравенство доказывается несколькими способами, может говорить о двух вещах:
либо Ваш результат очень глубокий, либо, наоборот, - довольно простой.
Интересно, это тот же Ваш друг, который фигурировал в следуюшем обсуждении?

arqady в сообщении #356302 писал(а):

Для неотрицательных $a$, $b$ и $c$ таких, что $ab+ac+bc\neq0$, докажите, что

$$\sqrt{\frac{a^3}{a^2+ab+b^2}}+\sqrt{\frac{b^3}{b^2+bc+c^2}}+\sqrt{\frac{c^3}{a^2+ac+c^2}}\geq\frac{\sqrt a+\sqrt b+\sqrt c}{\sqrt3}$$

У него имеется красивое доказательство.

ins- в сообщении #357003 писал(а):

A friend of mine said arqady's inequality can be solved easily by using Murhead.

arqady в сообщении #357105 писал(а):

Приведите доказательство вашего товарища, пожалуйста. Спасибо!

arqady в сообщении #357298 писал(а):

ins-, как на счёт доказательства с помощью Мюрхеда?

ins- в сообщении #357304 писал(а):

The person is under time pressure he have important project when he have more time he said he will post it.

Видимо, проект ещё не закончен...

[ins-]

"либо Ваш результат очень глубокий, либо, наоборот, - довольно простой." - most of the people say my result is simple. But simple is different for the different people. I know you are an inequalities expert and I saw you proved many many inequalities. What is your personal opinion about the inequality mentioned?

Yes, it is the same person. What makes me believe him I saw very hard problems he solved for example - hardest IMO problems and some bulgarian and vietnamese mo problems. The reason I suppose he didn't show me the solution is because he is very busy in writing software. He is an experienced software developer. I showed him our discussion and I believe when he have more time he will give an answer.

About my inequalities and geometry problems. I earn nothing from them. It is just curious for me to discover something new (or probably old but beautiful but not known to me) and to see it proven. People are different.

[arqady]

ins- в сообщении #555930 писал(а):

What is your personal opinion about the inequality mentioned?

Если Вы имеете в виду Ваше предыдущее неравенство, то я его доказал с помощью метода $SOS$. После нескольких довольно грубых оценок. Получилось очень громоздко, что говорит о том, что оно очень нестрогое. Простое доказательство, правда, не нашёл и поэтому своё громоздкое доказательство публиковать не хочу.
Если Вы имеете в виду неравенство из этого топика, то на вид оно ещё более громоздкое и поэтому лень мучиться ещё раз. Это всё имхо, конечно. Может, существует что-то красивое, а я это не вижу.

ins- в сообщении #555930 писал(а):

Yes, it is the same person... and I believe when he have more time he will give an answer.

Это за полтора-то года! Правда, для того чтобы сказать, что неравенство доказывается с помощью Мюрхеда, хватило и двух дней... Как-то это всё некрасиво выглядит, имхо.

[ins-]

http://www.artofproblemsolving.com/Foru ... 3#p2647243 - it is a beautiful proof at least for me. But I'm not an expert at all and it is a matter of taste.

What inspired me to "discover" these inequalities was problem 4. From the file on this link http://imomath.com/othercomp/Bul/BulMO497.pdf it also requires some calculations but looks very beautiful. I don't pretend my problems to be on the same level but I spent my time in a good for me way. The other source of inspiration for me was Vasile Cirtoaje's book where there are many beautiful inequalities.

I would like also to thank you for the time and effort spent on these inequalities and to excuse if I made you angry.

[ins-]

Does it help to put a=1/bc, b=1/ca, c=1/ab in all denominators in left and right expressions?

[ins-]

Dear arqady, and all the users interested in this topic. I would like to excuse me. This problem is not correct.
(0.8,1,1.25) is an example as an user discovered on the link below
http://www.math10.com/f/viewtopic.php?f ... 564#p45564

[ins-]

I'm sorry for the wrong information. After making some calculations in the mentioned by the user case my inequality is still true. I think he was wrong and the mentioned by me counter-example is not valid. Can someone verify this and to prove/disprove the inequality. It remained long time not proved/disproved. I suppose it is a hard inequality but probably correct.