Your transform is a logarithmic variation on the Young-Fenchel transform, which has an extensive literature, for example:
• On the Young-Fenchel transform for convex functions
• Variational Principles of Continuum Mechanics (chapter 5 on Young-Fenchel transformations)
More generally, one can define the Fenchel-Moreau 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.