It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual function $g$ which has the same properties, whereby the gradient is a bijection from the domain of $f$ to the domain of $g$.

Does there exist a similar set of results for strongly convex $f$? or for essentially strongly convex $f$?