Inequalities of Hardy type and generalizations on time scales

Abstract

In this paper, we prove some new dynamic inequalities on time scales which as special cases contain several generalizations of integral and discrete inequalities due to Hardy, Copson, Leindler, Bennett, Pachpatte and Pečarić and Hanjš.

Hardy–Copson type inequalities for nabla time scale calculus

“…We can refer to the surveys [1,33] and the monograph [2] for exhibition of these results. The aforementioned Hardy-Copson inequalities have been unified to an arbitrary time scale in the book in [3] and in the articles in [4,[34][35][36][37][38][39][40] by using delta time scale calculus.…”

“…The following theorem establishes the delta unification of discrete Bennett's inequality (1.6). Theorem 1.1 [39] For the functions w and f, let us define the functions A(t) = ∞ t w(s)∆s and…”

Higher integrability theorems on time scales from reverse Hölder's inequalities

“…The general idea is to prove the corresponding result for a dynamic inequality where the domain of a function is a so-called time scale T, which may be an arbitrary closed subset of real numbers R. These dynamic inequalities cover the classical continuous and discrete inequalities as special cases when T = R and T = N, and furthermore, they can be extended to different types of inequalities on various time scales such as T = hN, h > 0, T = q N for q > 1, etc. For this topic, the reader is referred to monographs [1,2], papers [3,17,18] and references therein. In particular, our aim in this paper is to extend the results due to D'Apuzzo and Sbordone [7], and Popoli [16], to time scales and derive the corresponding discrete results which will be essentially new.…”

“…Define Ψ(t) = b t ψ(x)∆x. Applying the integration by parts formula(17) to the term b a ϕ(t)Ψ(t)∆t with u(t) = Ψ(t) and υ ∆ (t) = ϕ(t), we get ∆ (t)υ σ (t)∆t, where υ(t) = t a ϕ(x)∆x. Now, since υ(a) = 0 and Ψ(b) = 0, it follows that )∆x ∆t, which completes the proof.…”

Hardy-type operators with general kernels and characterizations of dynamic weighted inequalities

“…For related dynamic inequalities, we refer the reader to the books [1,2] and the papers [8,10,33,32,34,35,36,39] and the references therein. Our aim in this paper is to prove some norm inequalities for a general operator of the form…”