Find all $f\in{C}^2(\mathbb{R})$ such that $(x,y)\rightarrow{f}(x+y)$ is subharmonic.

Question

I am trying to solve this problem:

Find all $f\in{C}^2(\mathbb{R})$ such that $(x,y)\rightarrow{f}(x+y)$ is subharmonic.

If $u(x,y)=f(x+y)$ is subharmonic then $\Delta{u}(x,y)\geq0$. This means that I want to find all $f(x)$ such that $\Delta{f}(x)\geq0$. But I don't know how to find more specific conditions.

Is this the right approach?

I would appreciate any kind of help.