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.
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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.
answered Apr 21, 2013 at 9:58
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$\begingroup$ Do you know any book in which one studies this logarithmic variation? $\endgroup$– AppliquéCommented Apr 21, 2013 at 10:21
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$\begingroup$ Maybe Rockafellar's book "Convex Analysis"? $\endgroup$ Commented Apr 9, 2014 at 18:57