This is a logarithmic variation on the Young-Fenchel transform, which has an extensive literature,:

<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.springerlink.com/content/978-3-540-88466-8">Variational Principles of Continuum Mechanics</A> (chapter 5 on Young-Fenchel transformations)