What does minorize mean?

Question

Let $f$ be closed and convex.

Then the conjugate of $f$ is $f^*(y) = \sup_{x}(y^Tx - f(x))$. If $y \in \text{dom}(f^*)$, then the affine function $h(x) = y^Tx - f^*(y)$ minorizes $f$.

What does minorize mean and how does $h$ minorize $f$?

Answer 1

"$f$ minorizes $g$" means $f \leq g$. In this case it's simple: $f^*(y) \geq y^T x - f(x)$ for all $x$ so $y^T x - f^*(y) \leq f(x)$ for all $x$.

"Majorize" is also a common term with the reverse meaning.

Answer 2

"Minorize" means "bound from below". So the assertion is just that $h(x) \le f(x)$ for all $x$.

Likewise, "majorize" means "bound from above".

I think these terms are more common among non-native English speakers; perhaps they are cognate with common terms in some other language?