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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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.