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This may help, if only partially: 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)

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.

This is a logarithmic variation on the Young-Fenchel transform, which has an extensive literature,:

On the Young-Fenchel transform for convex functions

Variational Principles of Continuum Mechanics (chapter 5 on Young-Fenchel transformations)

This may help, if only partially: 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)

I guess the reason you don't find references is because this is more commonly called the Fenchel transform or sometimes Young-Fenchel transform (and usually the logarithm is taken). There is an extensive literature, here are just a few pointers:

On the Young-Fenchel transform for convex functions

Continuity of the Fenchel transform of convex functions

This is a logarithmic variation on the Young-Fenchel transform, which has an extensive literature,:

On the Young-Fenchel transform for convex functions

Variational Principles of Continuum Mechanics (chapter 5 on Young-Fenchel transformations)