Inequality involving sum of fractions

Question

I'm looking for an inequality involving sums of fractions. In particular I'm looking for the inequality

$$ \frac{1}{n} \cdot \sum_{k=1}^n \frac{p_k}{q_k} \geq \frac{\sum_{k=1}^n p_i}{\sum_{k=1}^n q_k } $$ for positive values $p_i, q_i > 0$.

All examples that I tried, fulfilled the inequality, hence my question:

Does this inequality hold or is there an example that disproves it?

Answer 1

It's wrong.

Take $p_2=...=p_n\rightarrow0$.

We need to prove that: $$q_1+q_2+...+q_n\geq nq_1.$$ Now take $q_1\rightarrow+\infty.$

Answer 2

For $n=2$ we get $$(q_2p_1-q_1p_2)(q_2-q_1)\geq 0$$ Why should this be?