Kind of a disclaimer: I'm completely self-learning at the moment and need some feedback to stay on the right course.
Intro
So, I'm reading "Dynamic Pricing Without Knowing the Demand Function: Risk Bounds and Near-Optimal Algorithms" and can't figure out the meaning of the statement below that occurs on page 6 (in bold):
With regard to the demand function, we assume that λ(·) is non-increasing in the price p, has an inverse denoted by γ(·), and the revenue rate r(λ) := λγ(λ) is concave.
Making sense of notation
$\lambda(\cdot)$ is a function, and yet it is being given as an argument to the function $r(\cdot)$. Maybe $\lambda$ in that case is not a function but rather a result of that function, i.e. a resulting demand rate. Okay, let's roll with that.
Now, what was weird initially is how can revenue rate $r(\lambda)$ be concave? Like, at some point bigger demand causes lower revenue? Weird.
But then I realized that maybe $\lambda$ here is indeed the function $\lambda(\cdot)$, so $r(\lambda)$ is short for $r(\lambda(\cdot))$. In that case, indeed, the function would be concave (a negative parabola), since $r(\lambda(p)) = \lambda(p) * p$. Am I getting this right? EDIT: According to @Patricio, I am.
Revenue rate definition (and making sense of notation, still)
Another possible explanation is that I have a wrong idea of what a demand rate is (in the context of the Revenue Management literature). After all, no definition was given on the previous pages. Actually, no definition is given in the paper at all. And I haven't found anything useful on the Internet.
And that part — $r(\lambda) := \lambda\gamma(\lambda)$ — is a puzzle for me right now.
I haven't found information on that colon-equals sign apart from that it is used as an assignment operator in some programming languages.
Also, what is this mess (okay, I'm dramatizing...): $\lambda\gamma(\lambda)$? It's a function composition, right? But if it is, it should return the demand rate, no?
So, if what I've written in the "Making sense of notation" section is correct and it also is a function composition, then we can write it like this: $$r(\lambda(\cdot)) := \lambda(\gamma(\lambda(\cdot)))$$
Still, doesn't make sense to me. If $\gamma(\cdot)$ is inverse of $\lambda(\cdot)$, then $\gamma(\lambda(x))=x$. So, we get just $\lambda(\cdot)$ on the right side. Then $r(\lambda(\cdot)) = \lambda(\cdot)$?!
EDIT: I think I got it, although I need confirmation! It's actually both function composition and multiplication: $$r(\lambda(\cdot)) = \lambda(\cdot) * \gamma(\lambda(\cdot))$$ which, simplified, would be $r = q * p$.
$\lambda\gamma(\lambda)$ seems like a weird way to notate that but I'm a newbie, so I guess I have a lot to learn.
Conclusion
So, as you see, I'm confused. Really hope to get some feedback!