How to write the following function f(x,y) in CVX form

[Sun_Mao]

[Mark_L_Stone]

Can you show us the convexity proof? Perhaps (or perhaps not) that offers some insight as to how to formulate it for CVX.

it does appear to be convex, although I haven’t proven it. It would be neither convex nor concave if y rather than 1/y were multiplying the log term; and I don’t have intuition as to why 1/y would make it convex.

[Michal_Adamaszek]

Was discussed in the past. Convex and “looks conic”, but actually without a known conic representation.

How to express the function "1/(exp(x^2)-1)"

This t\geq \frac{1}{e^x-1},\quad x>0 is equivalent to 1\geq e^{-x}+\frac{1}{1+t} which can be written almost verbatim 1 >= exp(-x) + inv_pos(1+t) However your variant with x^2 seems much harder. Maybe you can reformulate the whole model somehow to avoid the square.

[Sun_Mao]

It is the proof

[Sun_Mao]

[Mark_L_Stone]

That is a non-constructive proof. So as per @Michal_Adamaszek 's post, you are out of luck unless someone comes up with something which they have not succeeded at in the past.