Inequality 1

If $a,b,c> 0$ , prove that $\sum_{\text{sym}} \frac{a^{2}}{b} \geq \frac{3\left(a^{3}+b^{3}+c^{3}\right)}{a^{2}+b^{2}+c^{2}} .$

#Algebra

Easy Math Editor

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Can you explain what is sym,pls.

Ferenets Roman - 1 year, 10 months ago

It means symmetry

You can write this thing as

$\frac { { a }^{ 2 } }{ b }+\frac { { b }^{ 2 } }{ c } +\frac { { c }^{ 2 } }{ a }+\frac { { a }^{ 2 } }{ c }+\frac { { b }^{ 2 } }{ a } +\frac { { c }^{ 2 } }{ b }$

Isaac YIU Math Studio - 1 year, 10 months ago

It's sum over all permutation of variables

Ryad M. - 1 year, 7 months ago

Would it not be $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a}$ only!

Nikola Alfredi - 1 year, 4 months ago

@Nikola Alfredi – Following it answer would be :

Proof: NOTE : All sums are symmetric in a,b,c.

$\sum \frac{a^2}{b} \geq 3\frac{\sum a^3}{\sum a^2}$

First we can see that $3\sum a^3 \geq 9abc$ , by AM-GM

Thus we need to show $\sum \frac{a^2}{b} \times \sum a^2 \geq 9abc$ .

Also, byAM-GM

$\sum a^2 \geq \sum ab$ Hence $\sum \frac{a^2}{b} \times \sum a^2 \geq \sum \frac{a^2}{b} \times \sum ab$

Thus by Cauchy-Schwarz Inequality :-

$\sum \frac{a^2}{b} \times \sum a^2 \geq \sum \frac{a^2}{b} \times \sum ab \geq (a^\frac{3}{2} + b^\frac{3}{2} + c^\frac{3}{2})^2$

and $(a^\frac{3}{2} + b^\frac{3}{2} + c^\frac{3}{2})^2 \geq 9abc$ by AM-GM

Thus $\sum \frac{a^2}{b} \times \sum a^2 \geq \sum \frac{a^2}{b} \times \sum ab \geq 9abc$ ...Q.E.D

If you face any problem in understanding my language or concept, feel free to ask.

Nikola Alfredi - 1 year, 4 months ago

@Nikola Alfredi – It is cyclic sum

Isaac YIU Math Studio - 1 year, 4 months ago

@Isaac Yiu Math Studio – Ok, I would search about it..

Nikola Alfredi - 1 year, 4 months ago

@Nikola Alfredi – You can use double cyclic sum

Ryad M. - 1 year, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$