Exponential function sum problem

[rustyjoker]

Homework Statement

I need to prove that
(1+x_{1})·...·(1+x_{n})≥(1-Ʃ^{n}_{i=1}x_{i}^2)e^{Ʃ^{n}_{i=1}x_{i}}
with all 0≤x_{i}≤1

I've already proven that

(1+x_{1})·...·(1+x_{n})≤e^{Ʃ^{n}_{i=1}x_{i}}
with all 0≤x_{i}≤1

and (1-x_{1})·...·(1-x_{i})≥1-Ʃ^{n}_{i=1}x_{i} with all 0≤x_{i}≤1 ,

but can't figure out what to do with the main problem :D

 

[sunjin09]

Homework Statement

I need to prove that
(1+x_{1})·...·(1+x_{n})≥(1-Ʃ^{n}_{i=1}x_{i}^2)e^{Ʃ^{n}_{i=1}x_{i}}
with all 0≤x_{i}≤1

I've already proven that

(1+x_{1})·...·(1+x_{n})≤e^{Ʃ^{n}_{i=1}x_{i}}
with all 0≤x_{i}≤1

and (1-x_{1})·...·(1-x_{i})≥1-Ʃ^{n}_{i=1}x_{i} with all 0≤x_{i}≤1 ,

but can't figure out what to do with the main problem :D

Take logarithm on both sides, i.e.,
Ʃ log(1+x_i)≥log(1-Ʃ x_i^2)+Ʃ x_i
then realize that x(1-x)≤log(1+x)≤x for 0<x<1, substitute in and you'll see

 

[rustyjoker]

you can't take logarithm because you can't know if 1-sum(x_i)^2 is negative or not.