Estimate the integral of the product of two functions

[nagornyi]

The functions g(x) and w(x) are defined on [a,b], the first one is bounded, the second one is positive bounded:
|g(x)|<=M
0<=w(x)<=m.

There are two additional conditions:
1. int[a,b] w(x) dx = 1
2. int[a,b] g(x) dx = 0

Question: | int[a,b] g(x) w(x) dx | <= ?
=============================

Using only the first condition, we can write:
| int[a,b] g(x) w(x) dx | <= |g(x)|*| int[a,b] w(x) dx | <= M.
Introduction of the second condition should (seems like) improve the estimate. But how?

 

[daon2]

Using only the first condition, we can write:
| int[a,b] g(x) w(x) dx | <= |g(x)|*| int[a,b] w(x) dx | <= M.
Introduction of the second condition should (seems like) improve the estimate. But how?

The second inequality is not true, or at least does not make much sense. Did you mean max[a,b]{|g(x)|} instead of |g(x)|?

See if this helps: we have for any real integral

\(\displaystyle \text{min}\{f(x)\}\cdot(b-a)\le \displaystyle \int_a^b f(x) dx \le \text{max}\{f(x)\}\cdot(b-a)\)

 

[nagornyi]

Thank you for the reply.

Did you mean max[a,b]{|g(x)|} instead of |g(x)|?

Yes, of course. The max it is.
As for the inequalities you suggested - not sure how to use them for the problem.