Sharp unifying generalizations of Opial’s inequality

Opial’s inequality and its ramifications play an important role in the theory of differential and difference equations. A sharp unifying generalization of Opial’s inequality is presented that contains both its continuous and discrete versions. This generalization, based on distribution functions, is extended to the case of derivatives of arbitrary order. This extension optimizes and improves the constant given in the literature. The special case of derivatives of second order is studied in more detail. Two closely related Opial inequalities with a weight function are presented as well. The associated Wirtinger inequality is studied briefly.