Boundedness of a function of two variables Let us consider the function from $\mathbb R^2$ to $\mathbb R$ defined by  $\phi(\alpha,\beta) = \frac{\alpha \beta}{\sqrt{\alpha ^4 + \beta ^4}}$ and $0$ otherwise.
If we look at the graph using Geogebra, then it is clearly bounded. However, proving this is the case feels like it requires a large number of cases, and it feels as though this is the incorrect approach.
It is trivial to show that when $\alpha = \beta$, then the function is bounded. Although from this point, it feels like a difficult process to bound the function in a simple way without introducing multiple cases for when $\alpha$ and $\beta$ are positive / negative / different and also when they are different combinations of $| \alpha | < 1$ and $| \beta | < 1$.
Is there a simpler solution to the problem, or would bounding this require all of the cases that I listed above?
 A: We use the fundamental inequality $ab\leq\frac{a^2+b^2}2$ for $a,b\geq0$.
We have
\begin{align*}
|\phi(\alpha,\beta)|&=\frac{|\alpha| |\beta|}{\sqrt{\alpha ^4 + \beta ^4}}=\frac{\sqrt{|\alpha|^2 |\beta|^2}}{\sqrt{\alpha ^4 + \beta ^4}}=\frac1{\sqrt{\alpha ^4 + \beta ^4}}\cdot\sqrt{|\alpha|^2 |\beta|^2}\\
&\leq\frac1{\sqrt{\alpha ^4 + \beta ^4}}\cdot\sqrt{\frac{|\alpha|^4+|\beta|^4}2}=\sqrt{\frac12},\ \ \ (\alpha,\beta)\neq(0,0).
\end{align*}
A: We may also rewrite
$$
\phi(\alpha,\beta) = \frac{\beta/\alpha}{\sqrt{1+(\beta/\alpha)^4}}.
$$
This is a function of one variable $t:=\beta/\alpha$, which should be amenable to standard elementary methods.
In fact, let $f(t)=\frac{t}{\sqrt{1+t^4}}$. Then
$$
|f(t)| = \frac{1}{\sqrt{t^{-2}+t^2}}.
$$
It is enough to show that $t^{-2}+t^2$ is bounded below by a positive constant, which reduces further to analyzing $u^{-1}+u$ for $u>0$. It's elementary to show that $u^{-1}+u$ has a global minimum (for $u>0$) at $u=1$ with value $2$. Backtracking all the way back, we may conclude that
$$
|\phi(\alpha,\beta)| = |f(\beta/\alpha)|\le \frac{1}{\sqrt{2}},
$$
with equality exactly when $\alpha=\pm\beta$.
(Of course, $\alpha=0$ or $\beta=0$ are special trivial cases not covered by the above).
