I guess the reason you don't find references is because this is more commonly called the Fenchel transform or sometimes Young-Fenchel transform (and usually the logarithm is taken). There is an extensive literature, here are just a few pointers:

<A HREF="http://www.ams.org/journals/proc/1988-104-04/S0002-9939-1988-0937844-8/">On the Young-Fenchel transform for convex functions</A>

<A HREF="http://www.ams.org/journals/proc/1986-097-04/S0002-9939-1986-0845984-5/">Continuity of the Fenchel transform of convex functions</A>