Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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WEIGHTED HARDY INEQUALITIES WITH SHARP CONSTANTS

  • Received : 2019.04.06
  • Accepted : 2019.11.06
  • Published : 2020.05.01
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Abstract

In the paper, we establish the validity of the weighted discrete and integral Hardy inequalities with periodic weights and find the best possible constants in these inequalities. In addition, by applying the established discrete Hardy inequality to a certain second-order difference equation, we discuss some oscillation and nonoscillation results.

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References

  1. A. Z. Alimagambetova and R. Oinarov, Criteria for the oscillation and nonoscillation of a second-order semilinear difference equation, Mat. Zh. 7 (2007), no. 1(23), 15-24.
  2. G. Bennett, Some elementary inequalities, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 152, 401-425. https://doi.org/10.1093/qmath/38.4.401
  3. G. Bennett, Some elementary inequalities. II, Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 156, 385-400. https://doi.org/10.1093/qmath/39.4.385
  4. P. Gao, Hardy-type inequalities via auxiliary sequences, J. Math. Anal. Appl. 343 (2008), no. 1, 48-57. https://doi.org/10.1016/j.jmaa.2008.01.024
  5. P. Gao, On $l^p$ norms of weighted mean matrices, Math. Z. 264 (2010), no. 4, 829-848. https://doi.org/10.1007/s00209-009-0490-2
  6. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1988.
  7. P. Hasil and M. Vesely, Critical oscillation constant for difference equations with almost periodic coeffcients, Abstr. Appl. Anal. 2012 (2012), Art. ID 471435, 19 pp. https://doi.org/10.1155/2012/471435
  8. P. Hasil and M. Vesely, Oscillation and non-oscillation criteria for linear and half-linear difference equations, J. Math. Anal. Appl. 452 (2017), no. 1, 401-428. https://doi.org/10.1016/j.jmaa.2017.03.012
  9. A. Kalybay, D. Karatayeva, R. Oinarov, and A. Temirkhanove, Oscillation of a second order half-linear difference equation and the discrete Hardy inequality, Electron. J. Qual. Theory Differ. Equ. 2017 (2017), Paper No. 43, 16 pp. https://doi.org/10.14232/ejqtde.2017.1.43
  10. A. Kufner, L. Maligranda, and L.-E. Persson, The Hardy inequality, Vydavatelsky Servis, Plzen, 2007.
  11. A. Kufner and L.-E. Persson, Weighted inequalities of Hardy type, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. https://doi.org/10.1142/5129
  12. L. Maligranda, R. Oinarov, and L.-E. Persson, On Hardy q-inequalities, Czechoslovak Math. J. 64(139) (2014), no. 3, 659-682. https://doi.org/10.1007/s10587-014-0125-6
  13. B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31-38. https://doi.org/10.4064/sm-44-1-31-38
  14. R. Oinarov, Oscillation of second order half-linear differential equation and weighted Hardy inequality, Mat. Zh. 12 (2012), no. 3, 149-155.
  15. B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, 219, Longman Scientific & Technical, Harlow, 1990.
  16. L.-E. Persson, R. Oinarov, and S. Shaimardan, Hardy-type inequalities in fractional h-discrete calculus, J. Inequal. Appl. 2018 (2018), Paper No. 73, 14 pp. https://doi. org/10.1186/s13660-018-1662-6
  17. P. Rehak, Oscillatory properties of second order half-linear difference equations, Czechoslovak Math. J. 51(126) (2001), no. 2, 303-321. https://doi.org/10.1023/A:1013790713905
  18. M. Vesely and P. Hasil, Oscillation and nonoscillation of asymptotically almost periodic half-linear difference equations, Abstr. Appl. Anal. 2013 (2013), Art. ID 432936, 12 pp. https://doi.org/10.1155/2013/432936