Abstract
In this paper, we establish some necessary and sufficient conditions for the validity of a generalized dynamic Hardy-type inequality with higher-order derivatives with two different weighted functions on time scales. The corresponding continuous and discrete cases are captured when mathbb{T=R} and mathbb{T=N}, respectively. Finally, some applications to our main result are added to conclude some continuous results known in the literature and some other discrete results which are essentially new.
Highlights
In [15] Hardy proved the classical continuous inequality∞1 x p f (t) dt dx ≤ p p∞ f p(x) dx0x 0 p–1 0 by using the calculus of variations in the twenties, where f (x) is a positive integrable function over any finite interval (0, x), f p is an integrable convergent function over (0, ∞), and p > 1
For more results on the study of inequalities of higher-order derivative, we refer the reader to the papers [21,22,23, 43, 44] and the references they cite
Following these trends and to develop the study of dynamic inequalities of Hardy-type of the differential forms on time scales, we prove the time scales version of the higher-order derivative inequality (1.8) on an arbitrary time scale T
Summary
In [15] Hardy proved the classical continuous inequality∞1 x p f (t) dt dx ≤ p p∞ f p(x) dx0x 0 p–1 0 by using the calculus of variations in the twenties, where f (x) is a positive integrable function over any finite interval (0, x), f p is an integrable convergent function over (0, ∞), and p > 1. In 1984 Gurka [11] proved that the following inequality, which contains the first-order derivative of u, b In [4, 9, 11] the authors studied some inequalities containing the first-order derivative with two different weighted functions.
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