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, positive-homogeneity 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.

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, positive-homogeneity 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.