Topics of studying inequalities $2$

After my first basic par here , in this note, I will try to emphasis on Cauchy Schwarz inequality. In my opinion, this inequality is one of the most widely used and the most elementry of all inequalities (here by elementry I do not mean easy, but it is root of many other inequalities). So here we start of with Cauchy Schwarz inequality.. First we will see a short form of this inequality. For any real $a,b,c,d$ we have $(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(bc-ad)^2$ So we get the inequality that $(a^2+b^2)(c^2+d^2)≥(ac+bd)^2$ The more generalized format can be written as $(a_1^2+a_2^2+...a_n^2)(b_1^2+b_2^2+....b_n^2)≥(a_1b_1+a_2b_2+......a_nb_n)^2$ The proof can be found here

Now, what type of problems can be designed on this topic, or else how to tackle these type on inequality. This problem is faced by a lot of people. So, whenever you see a problem, first try to relate it with AM-GM Inequality. Last note, I gave some good problems on AM-GM. Then Try to use Cauchy Schwarz . So, first Idea that I want to give a hint, based on my observations on various questions, I would like to add a tip here(Maybe it wont work all times). If ever you see a problem involving fractions i.e. Having $a,b,c,,,$ on both numerator and denominator, with the numerator a perfect square, then you can apply Cauchy Schwarz as such $\frac{a^2}{b}+\frac{c^2}{d}+.....≥\frac{(a+c+.......)^2}{b+d+....}$ The above is a direct applicaton of Cauchy Schwarz (Can you prove it?)

So, how Do we solve some inequalities involving Cauchy Schwarz ? Lets See

$EXAMPLE;1$ Show that for all positive $x,y,z$, We have $x+y+z≤2[\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}]$

$SOLUTION$:: Remember the point I raised in my notes. So, Now considering the right side of the inequality

$2[\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}]≥2(\frac{(x+y+z)^2}{2(x+y+z)}≥x+y+z$

So, we see that after a perfect application of Cauchy Schwarz Inequality, the problem remains Just 1 line proof question. So, lets see some more examples

$EXAMPLE;2$. First I would like to mention that this problem is taken from a Iran MO. So ,let us see the power of inequality in this case

Given that $x,y,z>1$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$, SO prove that

$\sqrt{x+y+z}≥\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}$

$Solution::$ So, how would you approach this question. Well the to prove part of inequality seems too difficult with AM-GM. Also in the right side of the inequality we have $x-1$ type of terms. SO we clearly see that simple product wont help. SO lets take a glance towards the given part. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$.

So, we try to get a term of $x-1$ type from given. So ,we clearly see that $\frac{x-1}{x}+\frac{y-1}{y}+\frac{z-1}{z}=1$

No take a look at the left part, there we have $x+y+z$ term. So see that from our above result if we try to apply $x×\frac{x-1}{x}=x-1$. SO now we get a clear idea to apply Cauchy Schwarz. Therefore, By Application Of Cauchy Schwarz Inequality, we get

$x+y+z=(x+y+z)(\frac{x-1}{x}+\frac{y-1}{y}+\frac{z-1}{z})≥(\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1})^2$

Which leads to

$\sqrt{x+y+z}≥\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}$

Now, Here are some beautiful problems that you must try in order to skill this topic. Also, if possible, I will be posting of more inequalities, Maybe Holder's Inequality or Jensen's Inequality of Just Advanced Cauchy Schwarz Inequality.

Problem $1$:: For $a,b,c,x,y,z$ Positive reals, prove that $(5ax+ay+bx+3by)^2\leq (5a^2+2ab+3b^2)(5x^2+2xy+3y^2)$

Problem $2$:: (Generalized) Let $a_1,a_2,.......a_n$ be positive reals. Prove that

$( 1 + a_1)(1+a_2) \ldots (1 + a_n) \geq ( 1 + \sqrt[n]{a_1 \times a_2 \times a_3 \times \ldots \times a_n})^n$

Problem $3$:: This is pretty good one. The solution( Of Mine) Looks very big and unsolvable, Bu dont stop bcz in end there yeilds easy simplification. This one is taken from KMO Winter Program Test . Prove that for $a,b,c>0$

$\sqrt{(a^2b+b^2a+c^2a)(ab^2+bc^2+ca^2)}≥abc+\sqrt[3]{(a^3+abc)(b^3+abc)(c^3+abc)}$

Apart from all these problems, you can also post your doubts or questions in the comments so as to proved more questions for more people too. Thanks :)