Is determining a non-constant solution to a functional inequality with polynomial arguements decidable?

Question

So suppose we have a functional inequality with polynomial arguments in $2$ variables, $\sum_i c_i f(p_i(x,y)) \geq 0$, where $c_i$ are say integer constants and $p_i$ are polynomials, say with integer coefficients. Is it a decidable problem to determine whether there are non-constant solutions for $f$? The motivation is that someone recently asked whether $f(x) + f(y) \geq f(x+y) + f(xy)$ had any non-constant solutions, and it stumped me but also made me wonder about whether a computer could figure it out.