Novel Diamond Alpha Bennett-Leindler Type Dynamic Inequalities and Their Applications

Abstract

For the exponent zeta > 1, the diamond alpha Bennett-Leindler type inequalities are established by developing two methods, one of which is based on the convex linear combinations of the related delta and nabla inequalities, while the other one is new and is implemented by using time scale calculus rather than algebra. These inequalities can be considered as the complementary to the classical ones obtained for 0 < zeta < 1. Since both methods provide different diamond alpha Bennett-Leindler type inequalities, we can obtain various diamond alpha unifications of the known delta and nabla BennettLeindler type inequalities. Moreover, the second method offers new Bennett-Leindler type inequalities even for the special cases such as delta and nabla ones. Moreover, an application of dynamic Bennett-Leindler type inequalities to the oscillation theory of the second-order half linear dynamic equation is developed and presented for the first time ever.