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

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.

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

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It is the proof

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.