Bennett–Leindler Type Inequalities for Nabla Time Scale Calculus

Diamond alpha Bennett‐Leindler type dynamic inequalities and their applications

“…The next theorem unifies the discrete Bennett-Leindler inequality (10) and the continuous Bennett-Leindler inequality (14) for arbitrary time scales and presents a nabla analogoue of the inequality (18). Theorem 8 (Kayar et al 52 ). For the functions w and f , we set…”

“…[35][36][37][38][39][40] For some results about the nabla differential equations and nabla inequalities, see previous works. [41][42][43][44][45][46][47][48][49][50][51] Contrary to delta case, Bennett-Leindler type inequalities had not been considered until the paper 52 appeared. The nabla time scale unifications of the discrete Bennett-Leindler inequalities (9) and (10) and the continuous Bennett-Leindler inequalities ( 13) and ( 14) as well as the nabla versions of Theorems 1-4 for an arbitrary time scale can be seen in the next theorems.…”

“…We notice that there is more than one way to obtain diamond alpha Bennett-Leindler type inequalities. Our first method is inspired from the papers 34,52 and is based on the convex linear combination of the aforementioned delta and nabla Bennett-Leindler type inequalities given in previous studies. 34,52 By this method, we establish Bennett-Leindler type integral inequalities whose left hand side consists of delta and nabla integrals and right hand side composed of diamond alpha integrals.…”

“…Section 2 presents the fundamental concepts of the time scale calculus in the sense of delta, nabla and diamond alpha derivatives and integrals. Section 3 is devoted to one of the main results, which is to unify the recently developed results presented in previous studies 34,52 into the diamond alpha time scale calculus by combining delta and nabla integral inequalities, both of which are special cases of 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 Theorems 1-8.…”

Diamond alpha Hardy-Copson type dynamic inequalities

“…The surveys [1,58] and the monograph [3] can be used to see these extended dynamic inequalities for delta approach. Although the nabla dynamic inequalities are less attractive compared to the delta ones, some of the nabla dynamic inequalities can be found in [6,13,27,40,41,53,56].…”