How to express a non-convex constraint in CVX?

How to express the non-convex constraint log(1+x) < log(1+1/y) in CVX?

The inequality is going in the wrong direction to be convex. Hence it can’t be used in CVX.

Presuming y is nonnegative. log(1+x) >= log(1+1/y) can be expressed with some reformulation in CVX, either with the logs still in place, or by removing them.