Does this inequality hold? $\sum \frac{a_i}{b_i} \leq \frac{\sum a_i}{\sum b_i}$

Question

I known that $\sum a_i b_i \leq \sum a_i \sum b_i$ for $a_i$, $b_i > 0$. It seems this inequality will also hold true when $a_i$, $b_i \in (0,1)$. However, I am unable to find out if

$\sum \frac{a_i}{b_i} \leq \frac{\sum a_i}{\sum b_i}$

holds true for $a_i$, $b_i \in (0,1)$.

Answer 1

Consider $$ \sum\frac{a_i}{b_i}\leq \frac{\sum a_i}{\sum b_i} \Leftrightarrow \sum b_i\sum\frac{a_i}{b_i}\leq\sum a_i. $$ Renaming $c_i=\frac{a_i}{b_i}>0$ implies $a_i=b_ic_i$ and $$ \sum b_i\sum\frac{a_i}{b_i}\leq \sum a_i\Leftrightarrow \sum b_i\sum c_i\leq \sum b_ic_i. $$

Answer 2

It doesn't take long to find a counterexample.

$$\frac11+\frac11>\frac22.$$

Note that the restriction to $(0,1)$ is immaterial as $\dfrac ab=\dfrac{ca}{cb}.$

Answer 3

Is it true that $\frac{2}{1}+\frac{3}{1} \le \frac{2+3}{1+1}?$

If you want numbers between $0$ and $1$, then consider $$\frac{0.2}{0.1}+\frac{0.3}{0.1} \ \ \le \ \ \frac{0.2+0.3}{0.1+0.1}$$

Answer 4

$a_1=0.1, a_2=0.2, b_1=0.3, b_2=0.4$ lead to the incorrect statement

$$\frac13 + \frac24 \le \frac37$$

In reality, the opposite inequality is true. You can see that if you rename $a,b$ to $x,y$ and rewrite it as

${\sum y_i} \sum \frac{x_i}{y_i} \geq \sum x_i$. This is the first inequality you said you knew, with $a_i:=y_i$ and $b_i:=\frac{x_i}{y_i}$,

Answer 5

Let $(a_n)$ be the sequence

$$1/2,1/4, 1/8,1/16,\dots,$$

and let $(b_n)$ be the same as $(a_n)$ except for $n=1,2,$ where let let $b_1=1/4, b_2= 1/2.$ Then $\sum a_n,\sum b_n$ both equal $1,$ hence so does their quotient, but

$$\sum \frac{a_n}{b_n} = \frac{1/2}{1/4} + \cdots > 2.$$