I think these tags (uvw and sos) are useful for the forum. User, which looks for to learn these methods, can click these tags and see many examples, how he can prove inequalities by these methods. For example. Let we need to prove that $$a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(...

I want to create a new tag EV-Method, but I want to know before, what Community thinks about it. This method is very useful for the proof of hard symmetric inequalities with $n$ variables. I think if our user will want to learn this method he'll can click this tag and see many examples, how to u...

I have been reading a book on some classic inequalties and i have stumbled upon this: Let $f:[0, +\infty]\rightarrow \mathbf R$ be a convex function and $x_1,x_2,...,x_n$ a sequence of positive numbers. It can be proved that :$$\sum_{i=1}^nf(x_i) \le (n- 1)f(0) + f(\sum_{i=1}^nx_i)$$ This is appa...

It is given that, $x+y+z=3\quad 0\le x, y, z \le 2$ and we are to maximise $x^3+y^3+z^3$. My attempt : if we define $f(x, y, z) =x^3+y^3 +z^3$ with $x+y+z=3$ it can be shown that, $f(x+z, y, 0)-f(x,y,z)=3xz(x+z)\ge 0$ and thus $f(x, y, z) \le f(x+z, y, 0)$. This implies that $f$ attains it's ...

Given $x,y,z$ are positive number satisfy $x^2+y^2+z^2=1$. Prove that $$\frac{x}{y^2+z^2}+\frac{y}{z^2+x^2}+\frac{z}{x^2+y^2}\ge \frac{3\sqrt{3}}{2}$$ i need a way use reduction of many fractions to a common denominator

$a, b,c $ are positive real numbers such that $a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$ Any ideas ?