Terminology for $f(tx) \leq tf(x)$ when $0\leq t \leq 1$ That $f(tx) \leq tf(x)$ for $0\leq t \leq 1$ is a famous property of convex functions, often used, for example, in the proof that convex functions are superadditive on the positive reals.
Edit: "A famous property of convex functions such that $f(0) \geq 0$", as the Wikipedia article mentions, sorry for not including that extra condition in my original post.
In economics, properties like $[f(tx) \le tf(x)$ for $0\le t \le 1]$ are often discussed in terms of "returns to scale".
My question is, in mathematics, is there a common name for the property $[f(tx) \le tf(x)$ for $0\le t \le 1]$?
 A: It is called a starshaped function, see definition 2 in [1].
Reference:
[1] A. M. Bruckner and E. Ostrow, "Some Function Classes Related to
the Class of Convex Functions", 1962.
A: There are convex functions that do not satisfy this property:
The functions $f(x)=x^2+a$ satisfy the property
$$\tag{1}
f(tx)\le tf(x)\,\quad\text{ for }0\le t\le 1
$$
only when $a\le 0\,.$ Indeed, $t^2x^2+a\le tx^2+at$ implies $a\le 0$ just by taking the case $t=0\,.$
In general it follows from (1) that $f(0)\le tf(0)\,,\,\forall t\in[0,1]\,.$
So that

*

*$f(0)\le 0\,.$
The property (1) is equivalent to
$$\tag{2}
\frac{f(x)}{x}\quad\text{ being increasing on } (-\infty,0)\text{ and on }(0,+\infty)\,.
$$
Let $0<y<x<\infty\,.$ Then $t=y/x\in(0,1)$ and (1) implies
$$
f(y)=f(tx)\le tf(x)=\frac{y}{x}f(x)\,.
$$
Therefore $f(x)/x$ is increasing on $(0,+\infty)\,.$ Let $-\infty<y<x<0\,.$ Then $t=x/y\in(0,1)$ and (1) implies
$$
f(x)=f(ty)\le tf(y)=\frac{x}{y}f(y)\,.
$$
Since $x<0$ this implies
$$
\frac{f(x)}{x}\color{red}{\ge}\frac{f(y)}{y}
$$
so that $f(x)/x$ is also increasing on $(-\infty,0)\,.$ Conversely, assume that (2) holds. If $x<0$ then for $0<t<1$ the point $y=tx$ is in $(x,0)\,.$ Then
$$
\frac{f(x)}{x}\le\frac{f(y)}{y}=\frac{f(tx)}{tx}\,.
$$
Because $x<0$ this implies
$$
f(tx)\le tf(x)\,.
$$
If $x>0$ then $y=tx$ is in $(0,x)$ and
$$
\frac{f(y)}{y}=\frac{f(tx)}{tx}\le\frac{f(x)}{x}
$$
which implies (1).
A: Such functions are said to be subhomogeneous. There’s a pattern here. If you have a name xxx for functions that satisfy an equality, then the name sub-xxx is used for functions that satisfy a similar inequality. So ….
Linear —> sublinear.
Additive —> subadditive.
Homogeneous —> subhomogeneous.
And so on.
I wouldn’t say the name subhomogeneous is common, but it’s not unknown, and it’s meaning is easy to guess.
