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Characterizations of reverse dynamic weighted Hardy-type inequalities with kernels on time scales

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Abstract

In this paper, we establish some conditions on nonnegative rd-continuous weight functions \(u\left( x\right) \) and \(\upsilon \left( x\right) \) which ensure that a reverse dynamic inequality of the form

$$\begin{aligned} \left( \int _{a}^{\infty }f^{p}(x)\upsilon \left( x\right) \Delta x\right) ^{ \frac{1}{p}}\le C\left( \int _{a}^{\infty }u\left( x\right) \left( \int _{a}^{\sigma \left( x\right) }\mathcal {K}\left( \sigma \left( x\right) ,\sigma \left( y\right) \right) f(y)\Delta y\right) ^{q}\Delta x\right) ^{ \frac{1}{q}}, \end{aligned}$$

holds when \(q\le p<0\) and \(0<q\le p<1.\) Corresponding dual results are also obtained. In particular, we prove some reverse dynamic weighted Hardy-type inequalities with kernels on time scales which as special cases contain some generalizations of integral and discrete inequalities due to Beesack and Heinig.

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Authors and Affiliations

  1. Mathematics Division, Faculty of Advanced Basic Sciences, Galala University, New Galala City, Egypt

    S. H. Saker

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

    S. H. Saker & M. M. Osman

  3. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    D. O’Regan

  4. Department of Mathematics, Texas A & M University, Kingsvilie, TX, 78363, USA

    R. P. Agarwal

Authors
  1. S. H. Saker
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  2. M. M. Osman
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  3. D. O’Regan
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  4. R. P. Agarwal
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Correspondence to S. H. Saker.

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Saker, S.H., Osman, M.M., O’Regan, D. et al. Characterizations of reverse dynamic weighted Hardy-type inequalities with kernels on time scales. Aequat. Math. 95, 125–146 (2021). https://doi.org/10.1007/s00010-020-00759-6

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