Many inequality problems on variable $x,y,z$ or $a, b$ come with constraints such as $ab = 1,xyz = 1, x+y+z = 1$. For example, Let $a$ and $b$ be positive real numbers with $a + b = 1$ then prove that - $$ \frac{a^2}{a + 1}+ \frac{b^2}{b + 1} ≥\frac1 3$$ (Hungary 1996).
Now I understand that contest-problem is good for exercise but what I am asking, is that are such inequality are useful in research? Or are such inequality found in research level mathematics?
This question is motivated by this post (click here).
By C-S we obtain: $$\frac{a^2}{a+1}+\frac{b^2}{b+1}\geq\frac{(a+b)^2}{a+1+b+1}=\frac{1}{3}.$$ Now we see that $\frac{1}{3}$ is a minimal value of the function $f(a,b)=\frac{a^2}{a+1}+\frac{b^2}{b+1}$
under conditions $a>0$, $b>0$ and $a+b=1$ (because the equality occurs for $a=b=\frac{1}{2}$).
Id est, we solved an optimization problem without standard Lagrange Multipliers method.
Specific this inequality is very easy, but of course it can be found in research level mathematics.