Let $a_{k},b_{k}>0$,show that $$\sum_{k=1}^{n}\dfrac{b_{k}(b_{1}+b_{2}+\cdots+b_{k})}{a_{1}+a_{2}+\cdots+a_{k}}<2\sum_{i=1}^{n}\dfrac{b^2_{i}}{a_{i}}\tag{1}$$
I known prove $b_{k}=1$,because it become $$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}<2\sum_{i=1}^{n}\dfrac{1}{a_{i}}$$ see mathstack
But How prove $(1)$?
