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

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 $$\frac{\sum_i a_i}{\sum_i b_i}$$.
• Titu's Lemma ${}$ Nov 14, 2020 at 13:51
• @TheSimpliFire No, it is not. Nov 14, 2020 at 16:39
• @Billy Rubina: What are you expecting? It's obvious after expanding Nov 14, 2020 at 16:42
• Are you looking for an upper bound or lower bound for $\sum \frac{a_i}{b_i}$? In the $a_1, a_2$ example you gave, its a lower bound that's shown. A lower bound is easy using CS inequality. Nov 14, 2020 at 17:15
• An upper bound isnt likely, just consider any one $b_i$, say $b_1 \to 0^+$. Nov 14, 2020 at 17:23

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 $$\frac{\frac 1N + \frac{N}{1}}{\frac{1+N}{N+1}} = N+\frac 1N > N.$$

• This is nice. My plan here was trying to find an upper bound for that quantity I judged harder to compute, do you think there is any other way to find a good upper bound for it? Dec 9, 2020 at 6:17
• Can't think of anything without additional information about the $a$'s and $b$'s. Dec 9, 2020 at 14:30
• What kind of information would yield nice results? Can you think on something? Dec 10, 2020 at 2:53
• Say if $m\le b_i \le M$ then $\sum a_i/b_i \le \frac Mm \frac{\sum a_i}{\sum b_i}$ Dec 10, 2020 at 15:00
• Thanks for your kind words. Back in high school and college I was lucky enough to have a few genius friends. I learned more from them than from the teachers/professors. Haven't studied any particular inequality books. I think you can just look around and find good peers to learn from. There are many experts in inequality here at Math SE as well. Good luck. Dec 11, 2020 at 1:31