"bounded from above" vs. "bounded above" For unary functions $f,g\colon X\to Y$, where $X$ is any set and $(Y,≤)$ is a poset, when you rephrase $\forall\,x{\in}X\colon f(x)≤g(x)$ in prose, do you write


*

*"$f$ is bounded above by $g$" or

*"$f$ is bounded from above by $g$" or

*"$f$ is majorized by $g$"?


Why?
(Remark: if we equip $X{\to}Y$ with the pointwise partial order, then $g$ is an upper bound for $f$ in this order, but this is not an answer to the question above.)
 A: (Too long for a comment)
In the book of Analysis I of Amann and Escher we can read in page 196:

Let $\sum x_n$ be a series in $E$ and $\sum a_n$ a series in $\Bbb R^+$. Then the series $\sum a_n$ is called a majorant (or minorant) for $\sum x_n$ if exists some $N\in\Bbb N$ such that $|x_n|\le a_n$ (or $|x_n|\ge a_n$) for all $n\ge N$.

what seem close to what we are searching for, so probably this terminology can be used over functions of any kind, not only series. But, in this case, the term is eventual, in the sense that it is not stated for all $n\in\Bbb N$, just for all $n\ge N$, for some $N\in\Bbb N$. 
And in Analysis I of Tao on page 317 we can read:

Let $f:I\to \Bbb R$ and $g:I\to \Bbb R$. Then we says that $g$ majorizes $f$ on $I$ if $g(x)\ge f(x)$ for all $x\in I$, and that $g$ minorizes $f$ in $I$ if $g(x)\le f(x)$ for all $x\in I$.

Then it seems we can use this definition of Tao, what is explicit about the topic of this question. However some definitions and terminology of Tao are not very common out of his books.
Moreover, we have an article of wikipedia about majorization, where it seems that the use of this terminology, in some context, is the same that what is discussed in this question.
By the other hand reading the article of wikipedia about big-O notation in this table we can read the concept of "bounded asymptotically". It seems that we can say that "$f$ is bounded above by $g$" with low risk to be misunderstood. 
