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> Prove $f(x, y) = \frac{x^2}{y}$ is a convex function on the set $\{(x,y) \in \mathbb{R}^2 : y > 0\}$.
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Attempt:

I start with the basic convexity, i.e.,
\begin{align}
f( \alpha\_1 x\_1 + \alpha\_2 x\_2) \leq \alpha\_1 f(x\_1) + \alpha\_2f(x\_2) \ ,
\end{align}
where $\alpha\_1 + \alpha\_2 = 1$.

Let $z = (x,y)$, then 
\begin{align}
f( \alpha z\_1 + (1-\alpha) z\_2) \leq \alpha f(z\_1) + (1-\alpha)f(z\_2) \ ,
\end{align}

How to proceed from here? Thank you so much.

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Question: Can this be proved by perspective function? If yes, how to prove that.