I just corrected a typo in my previous post. The corrected version is
More generally, x*log(1+y/x) = -rel_entr(x,x+y)
However the refomulation I showed for g1 was correct.
According to my calculations, 1/x(4)*log(1+a*x(3)/x(4)) is neither convex nor concave in (x(3) , x(4)). For instance, letting a=x(3)=x(4)=1, the Hessian has one positive and one negative eigenvalue. So this looks like a showstopper. There is no way to formulate g2 for CVX, because it is neither convex nor concave.
i don’t know what the paper did or showed, but that expression as is is not convex.
Your attempts to use CVX functions in disallowed ways will not succeed. You can’t sneak in disallowed argument types and convert a non-convex expression into a convex expression.