Is there a one-word term for "grows by square root"?

Question

We can describe a growth relationship $y = x$ as linear (e.g. "linear growth").

We can describe $y = \log(x)$ as logarithmic (e.g. "logarithmic growth").

We can describe $y = x^2$ as quadratic (e.g. "quadratic growth").

We can describe $y = 2^x$ as exponential (e.g. "exponential growth").

Is there a one-word term to describe a growth relationship where $y = \sqrt{x}$ ?

The best suggestion I've seen is "radical growth", which does not seem to be a standard term and I think would cause confusion if used without explanation.

HelloWorld · Accepted Answer · 2020-07-22T02:15:59.550

3

The asymptotic growth of a root is known as "sublinear growth". Another term is "fractional-power" growth but it sounds odd imo.

The sublinear term originates from linear algerba. It can be used not only for square-root but for any $n-th$ root complexity.

In strict terms as sublinear function is a function that satisfies the following properties:

Positive Homogeneity: $f(kx) = k^n f(x) \quad n\in\mathbb{R^+_0}$ and
Subadditivity: $f(x+y) \leq f(x) + f(y)$

Roots satisfy both. The proof of positive homogeneity is trivial. As far as the proof of subadditivity is concerned you can check this post

Other seemingly sublinear functions (such as log) do not satisfy these properties therefore, in a sense you can separate them from sublinears.

edited Jul 22 '20 at 02:15

answered Jul 21 '20 at 05:55

HelloWorld

1,564

1

My impression is that the term "sublinear" is also applicable to logarithmic growth; is that correct? I am hoping for a term that clearly distinguishes the growth as separate from logarithmic growth. – Kevin Jul 21 '20 at 06:07
@Kevin I edited my answer, to answer to your comment – HelloWorld Jul 21 '20 at 06:17
Roots do not satisfy "Positive Homogeneity". – BrianO Jul 21 '20 at 08:02
@BrianO $f(kx) = \sqrt k f(x)$ with degree of homogeneity $= \frac12 > 0$. e.g $\sqrt{kx} = \sqrt k \sqrt x$ – HelloWorld Jul 21 '20 at 08:45
@BrianO I edited my answer so its clearer. – HelloWorld Jul 21 '20 at 08:46
This can be generalized for n-th roots too btw – HelloWorld Jul 21 '20 at 08:50

score -1 · Answer 2 · answered Jul 21 '20 at 06:04

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Well since $y=\sqrt x \implies y^2=x$, I would call it inverse quadratic growth, i.e. the inverse of the function has quadratic growth.

answered Jul 21 '20 at 06:04

Rhys Hughes

12,842

We can also write it as $ y = x ^ {1/2} $ and call it "fractional quadratic growth", but that doesn't seem right either. – Kevin Jul 21 '20 at 06:13

Is there a one-word term for "grows by square root"?

2 Answers2