Is x/log(1+x) concave? How can I use cvx solve the constraints " x/log(1+x) >= C", where C is constant

[jackfsuia (Jack)]

Yes, when x>0, it’s cocave. Expressing it is equivalent to Expressing the following

How to formulate (1-exp(-x))/x

(1-\exp(-x))/x is convex in the whole real domain. I only need to use it in objective function for x>0. However, seems it’s hard to either certify it as convex using DCP or find a conic formulation with exponential cone or power cone. I suspect this is a case that’s out of range of DCP or common cones. Curious whether anyone know an answer.