Given a convex function $f$ from $R^2$ to $R$ which vanishes on $(0,1)$ and on $(1,0).$ Put $g(x,y)=1-x-y,$ where $(x,y) \in R^2.$ Which of these functions is less than the other $f(x,y)\leq g(x,y)$ or $f(x,y)\geq g(x,y)$ for $x,y$ in $]-\infty,0],$ $[0,1]$ and $[1+\infty[.$
Edit: $f$ is a decreasing function and $f$ is negative if $x,y > 1$ and $f$ is positive if $x$ and $y$ are $ < 1.$ Many thanks.