The Young transform of nonnegative function $f(x)$, $x \in \mathbb R^n_+$ is defined to be $$ (\mathscr Yf)(y) = \inf \left[ \left. \frac{x_1 y_1 + \ldots + x_n y_n}{f(x)} \; \right\; x \colon f(x) > 0 \right], \; y \in \mathbb R^n_+. $$ It preserves such properties as concavity, positivehomogeneity of first order, nonnegativity, continuity and arises in mathematical economics. It transforms the production function at the microlevel into the cost index of one unit of manufactured product. The problem is that I can't find anything about it in the internet. So any referencce to a book with study of this transform is very appreciated.
Your transform is a logarithmic variation on the YoungFenchel transform, which has an extensive literature, for example: • On the YoungFenchel transform for convex functions • Variational Principles of Continuum Mechanics (chapter 5 on YoungFenchel transformations) More generally, one can define the FenchelMoreau transform, $$(\mathscr F_{\phi}\;g)(y) = \inf_{x}\; \[g(x)\phi(x,y)], $$ with respect to a coupling function $\phi(x,y)$. The YoungFenchel 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. 

