# For positive reals $a_i$ and $b_i$, if $\sum a_i \geq \sum a_i b_i$, then $\sum a_i \leq \sum\frac{a_i}{b_i}$

Given $$a_1$$, $$a_2$$, $$a_3$$, $$\ldots$$, $$a_n$$ and $$b_1$$, $$b_2$$, $$b_3$$, $$\ldots$$, $$b_n$$ positive real numbers such that $$a_1 + a_2 + a_3 +\cdots+ a_n \geq a_1b_1+a_2b_2+a_3b_3...a_nb_n$$ show that: $$a_1 + a_2 + a_3+\cdots+a_n \leq \frac{a_1}{b_1} + \frac{a_2}{b_2} + \frac{a_3}{b_3}+\cdots+\frac{a_n}{b_n}$$

This was an admissions question for a math camp I ended up going to. After the program, I'm still wondering how to prove this. (The hint I got at the camp was to use Cauchy-Schwarz, but not sure where.)

I would love hearing solutions both with and without C-S, hope you can help!

• The given inequality can be rewritten as $a_1(1-b_1)+a_2(1-b_2)+...\geq0$. The second can be rewritten as $a_1(\frac1{b_1}-1)+a_2(\frac1{b_2}-1)+...\geq0$. $\frac1{b_1}-1=\frac{1-b_1}{b_1}$, so we seek to show that $a_1\cdot\frac{1-b_1}{b_1}+a_2\cdot\frac{1-b_2}{b_2}+...\geq0$. From here, you can apply Cauchy-Schwarz... Aug 14 '19 at 20:58
• Sorry, I'm still not clear where Cauchy-Schwarz comes into play Aug 14 '19 at 21:24
• Think about this: A given term in the desired inequality is negative iff $b_i>1$. If $b_i>1$, then $$|a_i\cdot\frac{1-b_i}{b_i}|<|a_i(1-b_i)|$$but clearly both are negative, so $$a_i\cdot\frac{1-b_i}{b_i}>a_i(1-b_i)$$If $b_i<1$, then clearly $$a_i\cdot\frac{1-b_i}{b_i}>a_i(1-b_i)$$ This means that if $b_i\neq1$, $$a_i\cdot\frac{1-b_i}{b_i}>a_i(1-b_i)$$If $b_i=1$, $$a_i\cdot\frac{1-b_i}{b_i}=a_i(1-b_i)=0$$ Aug 14 '19 at 22:04
• If we know that $$\sum_{k=1}^n a_k \geq \sum_{k=1}^n a_k b_k$$ why can't we immediately conclude that $$\sum_{k=1}^n \frac{a_k}{b_k} \geq \sum_{k=1}^n a_k$$ provided that $b_k \neq 0$? I had originally typed up an answer but realized that my reasoning was probably wrong as I didn't apply the CS inequality. Aug 15 '19 at 0:04

By C-S $$\sum_{k=1}^n\frac{a_k}{b_k}\sum_{k=1}^na_kb_k\geq\left(\sum_{k=1}^na_k\right)^2\geq\sum_{k=1}^na_k\sum_{k=1}^na_kb_k$$ and we are done!

• In order to form $$\sum_{k=1}^n\frac{a_k}{b_k}\sum_{k=1}^na_kb_k\geq\left(\sum_{k=1}^na_k\right)^2$$ Did you originally start at $$\sum_{k=1}^n\Big(\frac{a_k}{b_k}\Big)^2\sum_{k=1}^n\Big({a_k}{b_k}\Big)^2\geq\left(\sum_{k=1}^n (a_k)^2\right)^2$$ and take square roots? I'm curious due to the general form of C-S $$\bigg(\sum_{k=1}^n{a_k}^2\bigg)\bigg(\sum_{k=1}^n {b_k}^2\bigg)\geq\left(\sum_{k=1}^n a_k b_k\right)^2$$ Aug 15 '19 at 2:54
• @Axion004 I think it would be another problem. Aug 15 '19 at 3:40

I think this will work. I'm concerned that I didn't apply the CS inequality and therefore made a mistake near the last step.

$$a_1 + a_2 + a_3...a_n \geq {a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}...{a_n}{b_n}$$

can be rewritten as

$$a_1\Big(\frac{1-b_1}{b_1}\Big)+a_2\Big(\frac{1-b_2}{b_2}\Big)+\dots+a_n\Big(\frac{1-b_n}{b_n}\Big)\geq 0$$

then as $$a_i,b_i (i=1,2,\dots n)$$ are positive real numbers, we need to consider what happens to

$$a_i\Big(\frac{1-b_i}{b_i}\Big)$$

by analyzing three different cases.

Case 1: $$~0 < a_i,b_i < 1$$. In this case,

$$a_i\Big(\frac{1-b_i}{b_i}\Big) >a_i\Big({1-b_i}\Big)$$

since $$0 < b_i < 1$$.

Case 2: $$a_i=b_i=1$$. In this case,

$$a_i\Big(\frac{1-b_i}{b_i}\Big) =a_i\Big({1-b_i}\Big)=0$$

because $$b_i = 1$$.

Case 3: $$~a_i,b_i > 1$$. In this case,

$$a_i\Big(\frac{1-b_i}{b_i}\Big) >a_i\Big({1-b_i}\Big)$$

because both are negative but the LHS will be a smaller negative number.

This allows us to conclude that

$$\sum_{i=1}^n a_i\Big(\frac{1-b_i}{b_i}\Big) > \sum_{i=1}^n a_i\Big({1-b_i}\Big)$$

which can be rearranged to (provided $$b_i \neq 1$$ in which case the sum is $$0$$)

$$\sum_{i=1}^n \bigg(\frac{a_i\Big(\frac{1-b_i}{b_i}\Big)}{1-b_i}\bigg) > \sum_{i=1}^n a_i$$

so that

$$\sum_{i=1}^n a_i < \sum_{i=1}^n \Big(\frac{a_i}{b_i}\Big)$$

• What happens if a_i>1, a_j<1,b_k>1,b_l<1? Aug 15 '19 at 13:37