Is there any name for the fact that for any pair of finite sequences of real numbers $(a_i)_{i=1}^n$, $(b_i)_{i=1}^n$ there is $k \in [1,n]$ such that:
$$a_k \cdot \sum_{i=1}^n b_i \geq b_k \cdot \sum_{i=1}^n a_i$$
I know this is a generalization of the pigeonhole principle, and I know it's equivalent to Cauchy's mean value theorem in differential calculus, but it feels distinct enough to have a name of its own. Does it?
EDIT: To see the connection with pigeonhole principle, set $(b_i)$ to be constantly equal to 1. Then the equation can be rearranged:
$$a_k \ge \frac1n\sum_{i=1}^n a_i$$
And if $(a_i)$ is restricted to integers, it can be strengthened:
$$a_k \geq \lceil \frac 1n \sum_{i=1}^n a_i \rceil$$
I think it's clear enough now.
I will elaborate on the connection to Cauchy in an hour or so, as I have something to do and it requires careful constructions.
EDIT2: After careful consideration, the connection to Cauchy wasn't as strong as I thought. Instead, there's a fact that can be used to prove them both:
For any pair of integrable functions $f,g:[a,b] \rightarrow \mathbb R$ there is $\varepsilon \in [a,b]$ such that:
$$f(\varepsilon) \cdot \int_a^b g(x)dx \ge g(\varepsilon) \cdot \int_a^b f(x) dx$$
This can be seen from a very similar argument to what @Martin R used in the comment.
Now assume that both $f$ and $g$ are continuous and consider the function:
$$f(t) \cdot \int_a^b g(x)dx - g(t) \cdot \int_a^b f(x) dx$$
From the fact above, we can see that it has both a nonnegative value (the place where $f(t) \cdot \int_a^b g(x)dx$ prevails) and a nonpositive value (the place where $g(t) \cdot \int_a^b f(x) dx$ prevails). And since it's obviously continuous, by Darboux theorem it must be $0$ for some $\varepsilon \in [a,b]$, and that's what Cauchy's theorem asserts.
To see that the same fact proves the original fact, consider a class of functions on $[0,n]$ that are constant on every unitary sub-interval $(k,k+1], k \in \mathbb Z$. We can denote the values of this intervals as $a_1,...,a_n$. Even if someone knows very little of calculus, it is very easy to see that:
$$\int_a^b f(x) dx = \sum_{i=1}^n a_i$$
And from here the path is clear.
I originaly thought that the original fact proves this inequality on integrals, but my proof was flawed and I don't know if there is one.