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.

I don’t recall seeing this expression on the forum before.

Perhaps it’s a candidate for @Erling’s convex modeling challenge unless @Michal_Adamaszek, @hfriberg or another Mosek guru has a solution.

t \geq (1−\exp(−x))/x
is equivalent to
t \geq (\exp(y)-1)/y,\quad y = -x

and so matches the requested atom at How to express this convex function in CVX. We still haven’t found a DCP formulation in the currently supported set of cones. Out of curiosity, what are you trying to represent with (1−\exp(−x))/x (in what application does it appear)?

It showed up as average of exponential decaying.

You function seems to have to relationship to the Lambert function.