Abstract:
This paper is devoted to 0 < ζ < 1 complements of the classical (ζ > 1) diamond alpha Hardy-Copson type dynamic inequalities and their applications to dynamic equations. We obtain two kinds of diamond alpha Hardy-Copson type inequalities for 0 < ζ < 1, one of which is mixed type and established by the convex linear combinations of delta and nabla integrals while the other one is obtained by a new method which uses time scale calculus rather than algebra. In addition to their novelty, these two types are the complements of the classical diamond alpha Hardy-Copson type inequalities. In contrast to the works existing in the literature, these complements are derived by preserving the directions of the classical inequalities. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities obtained for 0 < ζ < 1 into one diamond alpha Hardy-Copson type inequalities and offer new types of diamond alpha Hardy-Copson type inequalities which have the same directions as the classical ones and can be considered as complementary to such inequalities. Moreover the application of these inequalities in the oscillation theory of half linear dynamic equations provides several nonoscillation criteria for
such equations.