# Tightness of an inequality

I have an inequality with $a_n>0\forall n$ and $A_n\geq a_n\forall n$ that \begin{equation} \sum^N_{n=1}a_n\frac{a_n}{A_n}\geq \frac{(\sum^N_{n=1}a_n)^2}{\sum^N_{n=1}A_n} \end{equation} however I am interested in how tight this inequality is. Does anyone have an idea to analyse it?

So far my idea is equality holds when \begin{equation} \frac{a_1}{A_1}=\cdots=\frac{a_N}{A_N} \end{equation} but any idea to show how much gap when the strict $>$ holds? For example, I tried to calculate the difference between two sides of inequality, but it becomes very complicated.

• It is sharp: Equality holds when $a_1 = \cdots = a_N = 0$. – Travis Oct 8 '14 at 10:55
• You should include the assumptions about the range allowed for $a_n,A_N$. For example, it appears that $A_n$ should be positive. Perhaps there are additional "known" restrictions that affect how "tight" the inequality is. – hardmath Oct 8 '14 at 10:55
• Yes you are right. I have edited the question. – user91232 Oct 8 '14 at 10:57

## 1 Answer

This is just the Cauchy–Schwarz inequality

https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality

It is well known (and follows from the proof) that equality occurs iff all numbers are proportional (this is the condition you wrote).