Proving the Inequality $1-x-\lambda \leq \frac{\prod_{i=1}^n (1+\lambda_i)}{\prod_{i=1}^n (1+x_i)}$

Question

I am trying to show the following inequality:

Let $\lambda_i \in [0,1]$ for $i=1,...,n$, and let $x_1,...,x_n$ be non-negative real numbers. Define $\lambda:=\sum_{k=1}^n \lambda_k$ and similarly $x:=\sum_{k=1}^n x_k$.

Then the following inequality holds:

$1-x-\lambda \leq \frac{\prod_{i=1}^n (1+\lambda_i)}{\prod_{i=1}^n (1+x_i)}$

My attempts: I don't really have any idea on how to approach this. I tried to do it by induction. But wasn't succesful. I tried to play around with the inequality to maybe get something more useful. But I get the feeling that there may be some specific trick for that inequality.

Answer 1

Enough is to prove that $(1-x)\leq\frac{1}{(1+x_1)\ldots (1+x_n)}.$ If $x>1$ this is obvious. If not we use $1+x_i<e^{x_i}$ leading to $$(1+x_1)\ldots (1+x_n)<e^x\leq \frac{1}{1-x}.$$