How the following function is bounded:
$f(x,y)=\frac{2xy^2}{(x^2+y^2)^2}$
My teacher wrote the above expression and told us that the expression is bounded. But if we see, then
$\frac{2y^2}{(x^2+y^2)^2}\leq 2$
Now, function can be written as:
$\frac{2|x|y^2}{(x^2+y^2)^2}\leq 2 |x|$
Clearly, $|x|$ is not bounded. So, the function should not be bounded. Is something wrong with my approach?.
I guess the intended meaning is a bounded function over $x^2+y^2\geq 1$, or over $x^2+y^2\geq\varepsilon$ for some $\varepsilon> 0$. Indeed, by the AM-GM inequality and Cauchy-Schwarz,
$$\left|xy^2\right|=\text{GM}(|x|,|y|,|y|)^3 \leq \left(\frac{|x|+2|y|}{3}\right)^3= \frac{1}{27}(x+2|y|)^3\leq \frac{1}{27}\left(\sqrt{5}\sqrt{x^2+y^2}\right)^3 $$ hence $$ \left|\frac{2xy^2}{(x^2+y^2)^2}\right|\leq \frac{10\sqrt{5}}{27}\cdot\frac{1}{\sqrt{x^2+y^2}}.$$