Test if function "raises faster then linear" We are given a noisy sample from a function $f(x)$, that can be approximated as some kind of a power function $f(x) \approx \mathcal{O}(x^p) + \varepsilon$, assuming $\varepsilon$ being homoscedastic, Gaussian noise. We want to test $H_0$: the increase of $f(x)$ is not faster then linear with $x$ ($p \le 1$), vs $H_1$: the increase of $f(x)$ is faster then linear ($p > 1$). What is the best way to conduct such test?
 A: Saying that a function “raises faster then linear” essencialy means that its derivative increases, meaning, its second derivative is positive.
The way you approximate the second derivative of a function is with a parabola. This is true for Taylor decomposition, when you want to approximate a function starting from a point evaluation of the function and its derivatives, but it works also for least squares. When fitting a straight line to your data, you are imposing a model with constant first derivative, but this can be amended adding a quadratic term, then the second derivative is constant, and you can allow it to vary adding a cubic term now, and so on.
But don't worry about how that (second) derivative varies, just settle with a mean estimate, it's the best thing you can use for testing.
When you consider a null model, that's the average $y$ value. When you have a linear model, the slope measures the average increment, when you include a quadratic term, that's the average second derivative. Simply test that for being positive.
A: Assuming you already know that $f$ is increasing, we can further posit that it increases super-linearly if its first derivative is monotone increasing in $x$ (this also makes it a convex function). Since we’re working with a discrete, countable set of observations
$$\{ (x_1, f_1) , (x_2, f_2), \dots, (x_n, f_n) \}$$
we can’t observe derivatives. But we can take a look at some form of discrete derivative, such as the forward difference of the series
$$\Delta f_i = f_{i+1} - f_i$$
for $i \in \{1, \dots, n-1\}$ (in this case, you'll have to discard the last observation $x_n$). Fitting a polynomial or a particular function by regressing $\Delta f_i$ on $x_i$ and checking the significance of the coefficients is not a robust solution since the functional form of the derivative can really take any non-polynomial shape. Also p-values of the regression coefficients aren’t accurate if there are significant departures from normality.
This is why I would instead recommend checking something like rank correlation between $\Delta f$ on $x$. Namely, Spearman correlation $\rho$ is a  non-parametric correlation based on rank which assess the monotonicity between two variables. And its statistical distribution is known both in small samples and large samples.
Thus, the one-sided test
$$H_0: \rho( \Delta f , x) = 0$$
$$H_A: \rho( \Delta f , x) > 0$$
if rejected, would lend credence to the claim that $f$ is indeed super-linear in $x$.
Numerical Example. Here, I'll generate two functions $f_0$ and $f_1$ with a $p$ of .8 and 1.2, respectively. Then I'll show that spearman correlation can distinguish which one is super-linear.
import numpy as np
from scipy.stats import spearmanr as sp # this is spearman correlation

delta = lambda series: series[1:] - series[:-1] # forward diff operator

n = 100 # size of sample
x = np.linspace(0,100,n) # x series
e = np.random.normal(0,1,n) # noise term
f0 = x**.8 + e # sub-linear function of x
f1 = x**1.2 + e # super-linear function of x


sp(delta(f0),x[:-1])

correlation=-0.034, pvalue=0.735


sp(delta(f1),x[:-1])

correlation=0.309, pvalue=0.002

While it doesn't invalidate the results of this experiment, keep in mind that to get accurate Type 1 error rate, that this p-value (from scipy) is for a 2-sided test. In your case, you are looking for a 1-sided test.
