How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $y \gt 0$? How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $ y \gt 0$?
I take the Hessian matrix of $\displaystyle \frac{x^2}{y}$, and I got:
$$H = \displaystyle\pmatrix{\frac{2}{y} & -\frac{2x}{y^2} \\-\frac{2x}{y^2} & \frac{2x^2}{y^3}}$$
further more, we have:
$$H = \frac{2}{y^3}\displaystyle\pmatrix{y^2 & -xy \\-xy & x^2}$$
I need to prove that H is semi-define positive matrix for y>0. I think I am close to the answer. But lack some knowledge to prove it since I try to calculate the $det(H)$ and it ends up equal to 0. Anyone can help? Thanks.

You're basically done: 
A symmetric matrix $(a_{ij})_{i,j\le n}$ is positive semidefinite iff the subdeterminants $\det(a_{ij})_{i,j\in H}$ are $\ge 0$ for all $H\subseteq\{1,2,..,n\}$.
