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)

More generally, one can define the <A HREF="http://www.encyclopediaofmath.org/index.php/Fenchel-Moreau_conjugate_function">Fenchel-Moreau</A> transform,

$$(\mathscr F_{\phi}\;g)(y) = -\inf_{x}\; \[g(x)-\phi(x,y)], $$

with respect to a coupling function $\phi(x,y)$. The Young-Fenchel transform corresponds to a bilinear $\phi$. Choosing $\phi(x,y)=\log(\sum_{n}x_n y_n)$ and $g(x)=\log f(x)$ gives essentially your transform.