Quantitative Borell-Brascamp-Lieb inequalities for compactly supported power concave functions (and some applications)

We strengthen, in two different ways, the so called Borell-Brascamp- Lieb inequality in the class of power concave functions with compact support. As examples of applications we obtain two quantitative versions of the Brunn- Minkowski inequality and of the Urysohn inequality for torsional rigidity.