Deciding on the correctness of an inequality

Assume $\forall i: x_i,y_i,a_i \ge 0$, and we know that $\forall i: \frac{a_i}{1+x_i} \ge \frac{a_i}{1+y_i}$ in other words $\forall i: x_i \le y_i$. I want to know if the following inequality

$$\sum_{i=1}^{n} \frac{a_i}{1+x_i} \ge \frac{\sum_{i=1}^{n}a_i}{1+\sum_{i=1}^{n}y_i}$$

is true? If the denominator of the RHS was $n + \sum_{i=1}^{n}y_i$ then it was easy to show it is true using properties of quotients, but for this one I'm not sure how to prove it, can't find a counter example either unfortunately, I would appreciate any hints.

BTW when I use Jensen's inequality (since $\frac{1}{x}$ is convex in non-negative orthant).

$$\sum_{i=1}^{n} \frac{a_i}{1+x_i} = \sum_{i=1}^{n} \frac{\frac{a_i}{\sum_{j=1}^{n}a_j}}{1+x_i}\sum_{j=1}^{n}a_j \ge \frac{\sum_{j=1}^{n}a_j}{1 + \sum_{i=1}^{n}\frac{a_ix_i}{\sum_{j=1}^{n}a_j}}$$

How can we proceed from there?

Solution 1:

Denote by $s_y = \sum\limits_{i = 1}^n y_i$ the sum of the $y_i.$ Since the $y_i \geq 0,$ we have $s_y \geq y_i$ and then $$ \dfrac{a_i}{1 + x_i} \geq \dfrac{a_i}{1 + y_i} \geq \dfrac{a_i}{1 + s_y}. $$ Summing over $i$ gives the desired result.

PS: I've been in similar situations where using the same index $i$ on two different sum inside the same expression makes me visually believe they are connected, I suspect this was your issue too.