I have a function of the form $f(x,y) = \frac{x}{c+y}$ where $c$ is a positive constant and $d \ge x, y \ge 0$. I would like to find a convex upper-bound for this function. Is there a principled way for doing this? How about if the upper-bound has to be convex and piecewise linear? Is there a way to find the optimal upper-bound in terms of the number of linear pieces?

I have a function of the form $f(x,y) = \frac{x}{c+y}$ where $c$ is a positive constant, $c \ge x \ge 0$, and $y \ge 0$. I would like to find a convex upper-bound for this function. Is there a principled way for doing this? How about if the upper-bound has to be convex and piecewise linear? Is there a way to find the optimal upper-bound in terms of the number of linear pieces?

Update: Sorry I changed the boundary conditions on $x$ and $y$ so it suits my problem better.

I have a function of the form $f(x,y) = \frac{x}{c+y}$ where $c$ is a positive constant and $d \ge x, y \ge 0$. I would like to find a convex upper-bound for this function. Is there a principled way for doing this? How about if the upper-bound has to be convex and piecewise linear? Is there a way to find the optimal upper-bound in terms of the number of piecewise linear terms?

I have a function of the form $f(x,y) = \frac{x}{c+y}$ where $c$ is a positive constant and $d \ge x, y \ge 0$. I would like to find a convex upper-bound for this function. Is there a principled way for doing this? How about if the upper-bound has to be convex and piecewise linear? Is there a way to find the optimal upper-bound in terms of the number of linear pieces?