Hardy–Copson type inequalities for nabla time scale calculus

Abstract

This paper is devoted to the nabla unification of the discrete and continuous Hardy-Copson-type inequalities. Some of the obtained inequalities are nabla counterparts of their delta versions while the others are new even for the discrete, continuous, and delta cases. Moreover, these dynamic inequalities not only generalize and unify the related ones in the literature but also improve them in the special cases.

Diamond alpha Hardy-Copson type dynamic inequalities

“…Contrary to delta case, nabla Hardy-Copson type inequalities have not been considered until 2021. The first results of this case were obtained by Kayar and Kaymakçalan in [33].…”

“…Although the authors presented Theorem 1.1-Theorem 1.3 in [64] for delta time scale calculus, they did not include the following theorem. For the completeness of the paper, we give the next theorem, which is a delta unification of the Hardy's discrete inequality (1.1), the Copson's discrete inequality (1.4) (or the Bennett's discrete inequality (1.7)) and of the continuous inequalities (1.2), (1.3), (1.10) as well as of their continuous generalization (1.12), established in [33].…”

“…Delta, nabla and diamond alpha time scale calculi and their main propoerties are introduced in Section 2. The contribution of Section 3, which includes one of the main results, is to unify the recently developed results presented in [33,64] for time scale diamond alpha calculus by combining dynamic delta and nabla integral inequalities, both of which are special cases of dynamic diamond alpha integral inequalities. Another main result of this paper is given in Section 4 which contains diamond alpha unifications of the foregoing dynamic delta and nabla integral inequalities proven in Theorem 1.1-Theorem 1.8.…”

“…We notice that there is more than one way to obtain diamond alpha Hardy-Copson type inequalities. Our first method is inspired from the papers [64] and [33] and is based on the convex linear combination of the aforementioned delta and nabla Hardy-Copson type inequalities given in [33,64]. By this method, we establish Hardy-Copson type integral inequalities whose left or right hand side consists of delta and nabla integrals and right or left hand side composed of diamond alpha integrals, respectively.…”

“…The growing interest to Hardy-Copson type inequalities have taken place in the time scale calculus as well and delta unifications of these inequalities have been established in the book [4] and in the articles [59]- [65], [2,17,18,22,23] whereas their nabla counterparts and extensions can be seen in [33][34][35].…”