I believe x*log(1+x/y) is convex for x > 0, y > 0.
I don’t see how to get it into CVX. Perhaps someone else knows how, although I don’t know whether it can be done.
As has been posted on this forum, the “similar” looking x*log(1+y/x) can be expressed as -rel_entr(x,x+y), and is concave for x > 0, y > 0. But I don’t see how to use rel_entr for x*log(1+x/y)