This afternoon I wrote a brief post about Terence Tao’s new paper A Maclaurin type inequality. That paper builds on two classical inequalities: Newton’s inequality and Maclaurin’s inequality. The previous post expanded a bit on Newton’s inequality. This post will do the same for Maclaurin’s inequality.

As before, let *x* be a list of real numbers and define *S*_{n}(*x*) to be the average over all products of *n* elements from *x*. Maclaurin’s inequality says that *S*_{n}(*x*)^{1/n} is decreasing function of *n*.

*S*_{1}(*x*) ≥ *S*_{2}(*x*)^{1/2} ≥ *S*_{3}(*x*)^{1/3} ≥ …

We can illustrate this using the Python code used in the previous post with a couple minor changes. We change the definition of `ys`

to

ys = [S(xs, n)**(1/n) for n in ns]

and change the label on the vertical axis accordingly.

Looks like a decreasing function to me.