Differential inequalities associated with weighted symmetrization processes on the real line

We consider the differential operator in a symmetric intervalJ ofR, where ϕ is the density function of a finite measure supported inJ satisfying a certain regularizing property. Using as main tools left symmetrization with respect to the weight ϕ, and certain measure preserving transformations, we prove differential inequalities involving the differential operators . As applications, we give comparison results for solutions to non-linear differential equations related to the operator, and with Neumann boundary data, generalizing in different ways results by C. Borell in Bor 85b]. Research supported in part by a Grant from El Ministerio de Educación y Cultura (Spain) PB97-1081, and from La Junta de Andalucía.