Diamond-Alpha Hardy-Copson Type Dynamic Inequalities-Ii

Abstract

The standard diamond-alpha Hardy-Copson type dynamic inequalities, which are derived for p > 1, are complemented to the new case p < 0. The novel approach for these complements is that the directions of the standard inequalities remain same while these complements are established by using a new method. This method does not require the Integration by Parts Formula and the Fundamental Theorem of Calculus, which are the standard arguments in the proofs of the similar theorems in the delta and nabla approaches but do not exist in the diamond-alpha calculus. Since it is the first time for analyzing the case p < 0, these complementary inequalities are novel not only for the diamond alpha calculus but also for the special cases, which are delta, nabla, continuous and discrete cases. © 2025 selection and editorial matter, Ravi P. Agarwal, Bipan Hazarika and Sanket Tikare. All rights reserved.