Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Exercise 1.4 from a great book The Cauchy-Schwarz Master Class asks to prove the following:

For all positive $x$, $y$ and $z$, one has $$x+y+z \leq 2 \left(\frac{x^2}{y+z} + \frac{y^2}{x+z} + \frac{z^2}{x+y}\right).$$

Introduction to the exercise says:

There are many situations where Cauchy's inequality conspires with symmetry to provide results that are visually stunning.

How to prove that inequality? And how does one benefit from the "symmetry"? What is the general idea behind this "conspiracy"?

share|improve this question
6  
The back of the book has solutions to the exercises. –  Byron Schmuland Oct 10 '11 at 19:09
1  
(This was my original comment, that I removed by mistake.) I don't think the answers here will explain the conspiracy better than Steele does in his book. (But let's hope that I am wrong about this.) So my recommendation is: read the book! :) –  Srivatsan Oct 10 '11 at 19:43
    
Uh, I didn't even realize that the book has solutions. I've only printed out the first chapter, but the book is definitely worth buying. Unfortunately, as Martin has noticed, the back of the book doesn't elaborate on the conspiracy. –  JoeCamel Oct 10 '11 at 21:10

5 Answers 5

$(x+y+z)^2 = \left(x\sqrt{\frac{y+z}{y+z}} +y\sqrt{\frac{x+z}{x+z}}+z\sqrt{\frac{x+y}{x+y}}\right)^2\leq\left(\frac{x^2}{y+z} + \frac{y^2}{x+z} + \frac{z^2}{x+y}\right)(y+z+x+z+x+y)$. You will get it. I didn't read that book, but I believe Srivatsan is right.

share|improve this answer

The inequality is $$f\left(\frac{x+y+z}{3}\right) \leq \frac{f(x)+f(y)+f(z)}{3}$$ for all positive $x,y,z$ with sum $A$, where $f(u) = \frac{u^2}{A-u}$, and $A=x+y+z$.

This is equivalent to $f(u)$ being convex in the interval $(0,A)$. The second derivative is $f''(u) = \frac{2A^2}{(A-u)^3} > 0$.

share|improve this answer

The inequality $$ (x+y+z)^2\le \left(\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}\right)[(y+z)+(x+z)+(x+y)]$$ can be obtained from Cauchy-Schwarz.

(You use it for triples $\frac{x}{\sqrt{y+z}}$, $\frac{y}{\sqrt{x+z}}$, $\frac{z}{\sqrt{x+y}}$ and $\sqrt{y+z}$, $\sqrt{x+z}$, $\sqrt{x+y}$.)

Now if you cancel $(x+y+z)$ you get the desired inequality.


However, I am not able to explain "the conspiracy", so I left this task for other answerers.


After seeing Byron Schmuland's comment I had a look into the book, and found out that basically the same explanation is given in the back of the book.

share|improve this answer

I have solved it without using Cauchy's Inequality

We have to prove that

$\dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{y+x}=(x+y+z)\left(\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{y+x}-1\right)\geq \left(\dfrac{x+y+z}{2}\right)$

$\implies \dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{y+x}\geq \dfrac{3}{2}$

And this is Nesbitt's Inequality (The proof is given in this link http://en.wikipedia.org/wiki/Nesbitt's_inequality)

Hence Proved.

share|improve this answer

There is an inequality which can be deduced by using the Cauchy Schwarz inequality:

Let $a_1,...,a_n$ be real numbers and $b_1,...,b_n>0$. Then $$ \sum_{i=1}^n \frac{a_i^2}{b_i} \geq \frac{(a_1+...+a_n)^2}{b_1+...+b_n}$$

The desired inequality is a simple consequence of the above.

share|improve this answer
1  
This is called Begstrom inequality. –  Sunni Oct 10 '11 at 20:38
    
Bergström, not Begstrom. –  Byron Schmuland Jan 8 at 18:34

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.