This may help, if only partially: Your transform is a logarithmic variation on the <A HREF="http://www.encyclopediaofmath.org/index.php/Dual_functions">Young-Fenchel transform,</A> which has an extensive literature, for example:

&bull; <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>

&bull; <A HREF="http://www.springerlink.com/content/978-3-540-88466-8">Variational Principles of Continuum Mechanics</A> (chapter 5 on Young-Fenchel transformations)