How can we find a good upper bound for $\sum_i \frac{a_i}{b_i}$ in terms of $\frac{\sum_i a_i}{\sum_i b_i}$

Question

I've been thinking about inequalities of the kind:

$$\frac{\sum_i a_i}{\sum_i b_i}\leq \sum_i \frac{a_i}{b_i} \qquad a_i,b_i>0$$

For some cases, it is easy to see why it works, for example:

$$\frac{a_1+a_2}{b_1+b_2}\leq \frac{a_1}{b_1}+\frac{a_2}{b_2}\qquad a_i,b_i>0$$

We write

$$0\leq a_1b_2^2+a_2b_1^2$$

And then:

$$a_1b_1b_2+a_2b_1b_2 \leq a_1b_1b_2+a_2b_1b_2+ a_1b_2^2+a_2b_1^2$$

Finally:

$$b_1b_2(a_1+a_2)\leq (a_1b_2+a_2b_1)(b_1+b_2)$$

Which yields the desired inequality:

$$\frac{a_1+a_2}{b_1+b_2}\leq \frac{a_1}{b_1}+\frac{a_2}{b_2}$$

I am curious about the following:

• How can we find a good upper bound for $\sum_i \frac{a_i}{b_i}$ in terms of $\frac{\sum_i a_i}{\sum_i b_i}$? In my hypothetical application for this, obtaining the number $\frac{\sum_i a_i}{\sum_i b_i}$ is very easy but obtaining $\sum_i \frac{a_i}{b_i}$ would be very messy. Having a good upper bound in terms of $\frac{\sum_i a_i}{\sum_i b_i}$ would be desirable. I know next to nothing about inequalities. I guessed about squares, cubes of $\frac{\sum_i a_i}{\sum_i b_i}$ but this would perhaps be too large. I guess I don't know how to find better upper bounds in terms of $\frac{\sum_i a_i}{\sum_i b_i}$.

Answer 1

If you want an upper bound that only depends on $\frac{\sum a_i}{\sum b_i}$ then no.

The quantity $\frac{\sum a_i}{\sum b_i}$ is called the mediant, or freshman sum, of the fractions $\frac{a_i}{b_i}$. It is a weighted average of the fractions so it's always between $\min \frac{a_i}{b_i}$ and $\max \frac{a_i}{b_i}$ and can be arbitrarily close to either. And this is exactly how Simpson's Paradox works.

https://en.wikipedia.org/wiki/Mediant_(mathematics)

Now if $n=2$, WLOG assume $\frac{a_1}{b_1} < \frac{a_2}{b_2}$. The mediant can be arbitrarily close to $\frac{a_1}{b_1}$ while the ratio of $\frac{a_2}{b_2}$ to $\frac{a_1}{b_1}$ can also be arbitrarily large. Therefore $\frac{\sum \frac{a_i}{b_i}}{\frac{\sum a_i}{\sum b_i}}$ can be arbitrarily large.

For example, if $N$ is very large, then $$ \frac{\frac 1N + \frac{N}{1}}{\frac{1+N}{N+1}} = N+\frac 1N > N. $$