Proving $f(x,y) = |xy| + a(x^2 + y^2)$ is convex for $a \ge 1/2$ I am working on a problem to prove $$
f(x,y) = |xy| + a(x^2 + y^2)
$$
is convex for $a \ge 1/2$.
My approach is to show that the Hessian is PSD for the cases where $x \not= 0, y \not= 0$.
However, this approach breaks down for cases when $x$ or $y$ are zero since the function is not differentiable at these points.
Any suggestions on other approaches that get around this?

$\begin{array}\\
f(x,y) 
&= |xy| + a(x^2 + y^2)\\
&= |xy| + (a-\tfrac{1}{2})(x^2+y^2)+\tfrac{1}{2}|x|^2+\tfrac{1}{2}|y|^2\\
&= (a-\tfrac{1}{2})(x^2+y^2) + \tfrac{1}{2}(|x|+|y|)^2\\
\end{array}
$
The first term, $(a-\tfrac{1}{2})(x^2+y^2)$, is clearly convex.
The second term is the composition of the (outer) convex increasing nonnegative function $\xi\mapsto \tfrac{1}{2}\max\{0,\xi^2\}$ and the inner function $(x,y)\mapsto |x|+|y|$. 
