Is this inequality true and provable?

3029298

1. The problem statement, all variables and given/known data
My question is whether the following inequality can be proven.


2. Relevant equations
[tex]
\left|\int_a^bg\left(x\right)dx-\int_a^bh\left(x\right)dx\right|\leq\int_a^b\left|g\left(x\right)-h\left(x\right)\right|dx
[/tex]


3. The attempt at a solution
I tried to write down the inequality in the form of it's primitives, where [tex]G\left(x\right)[/tex] is the primitive of [tex]g\left(x\right)[/tex] and [tex]H\left(x\right)[/tex] is the primitive of [tex]h\left(x\right)[/tex]. The inequality then becomes:

[tex]
\left|G\left(b\right)-G\left(a\right)-H\left(b\right)+H\left(a\right)\right|\leq\left|G\left(b\right)-H\left(b\right)\right|-\left|G\left(a\right)-H\left(a\right)\right|
[/tex]

But what next, or are there other means of getting a proof?

snipez90

Assuming [tex] a \leq b [/tex] and f is continuous on the interval [a,b], then

[tex] \left|\int_a^bf\left(x\right)dx \right| \leq \int_a^b\left|f(x)\right|dx[/tex]

which follows from the fact that [tex] f(x) \leq \left|f(x)\right| [/tex] and [tex] -f(x) \leq \left|f(x)\right| [/tex] and that

If f,g are both continuous on the interval [a,b] and [tex] f(x) \leq g(x) [/tex] for all x in the interval. Then

[tex] \int_a^b f(x)dx \leq \int_a^b g(x)dx [/tex]

Rearranging and using the first inequality should give you the desired inequality.

3029298

Oh, I see it now, it is indeed not that difficult.

[tex]
\left|\int_a^bg\left(x\right)dx-\int_a^bh\left(x\right)dx\right|\leq\int_a^b\left|g\left(x\right)-h\left(x\right)\right|dx
[/tex]

If we rearrange:

[tex]
\left|\int_a^b\left(g\left(x\right)-h\left(x\right)\right)dx\right|\leq\int_a^b\left|g\left(x\right)-h\left(x\right)\right|dx
[/tex]

Substituting [tex]f\left(x\right)=g\left(x\right)-h\left(x\right)[/tex] and using the first formula of snipez90, we get the proof.

Thanks!