A convex function, with one secant line drawn in red.

Haven’t had a straight up “math” day in quite a while here. That ends *now*. While reading an article [PDF link], I came across the definition of a *Young function:* a function which is convex and where *F(t) = 0* only when *t = 0*. Recall that for a real valued function, *convex *means that every secant line lies above the curve. So far, so good. Such a function might serve as a measure of how alike two functions are: rather than looking at (for example) the norm of *u-v*, we might first see what we can say about *F(|u-v|)*, a more general question.

But here’s the proposition that made me pause: For *1<q<m<p*, there is a differentiable Young function *F* and a constant *C(m,p,q)>0* so that

1. , and

2. for all *t > 0*.

A non-convex function, with a secant line drawn in passing beneath the function.

In fact, there was one more somewhat technical, but non-trivial assertion about this *F* (proposition 2.1 in the linked paper), but let’s focus on these two. Initially I was convinced that no such *F *existed, even satisfying these two conditions. Here’s how my erroneous thoughts went: suppose such an *F* were to exist. Property 2 gives us then that . Solving this “differential inequality” gives us that . A similar calculation will also yield that .

Now as a “back of the envelope” calculation, I tried plugging the bounds of *F* into property 1. Specifically, first I computed

.

Since *q<m*, the exponent is (strictly) smaller than 1, and the integral diverges (the indefinite integral looks like , where ). In particular, it does great from 0 to any finite number, but has a “fat tail”. Similarly, the integral diverges, but this time because its singularity near zero is too big (the indefinite integral is the same as the previous one, though now $\alpha < 0$. So this one does great from a very small number to infinity, but ultimately diverges.

Possibly I’ve given away the answer since I have emphasized my mis-steps, but here is an example of a Young function satisfying the first two properties. The trick is to construct the function out of two pieces:

The constructed Young function, near where the two "pieces" join.

for small *t, *and for large *t.* You can even select so that the derivative is continuous. Explicitly, suppose *m = 3*. Then we may set *F(t) = t* for *t < 1*, and for . Notice that *F(t) *is continuously differentiable, and that , so we know that .

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