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On Hardy q-inequalities

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Abstract

Some q-analysis variants of Hardy type inequalities of the form

$${\int_0^b {\left( {{x^{\alpha - 1}}\int_0^x {{t^{ - \alpha }}f(t){{\text{d}}_q}t} } \right)} ^p}{{\text{d}}_q}x \leqslant C\int_0^b {{f^p}(t){{\text{d}}_q}t} $$

with sharp constant C are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.

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References

  1. W. A. Al-Salam: Some fractional q-integrals and q-derivatives. Proc. Edinb. Math. Soc., II. Ser. 15 (1966), 135–140.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Bangerezako: Variational calculus on q-nonuniform lattices. J. Math. Anal. Appl. 306 (2005), 161–179.

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Bennett: Factorizing the Classical Inequalities. Memoirs of the American Mathematical Society 576, AMS, Providence, 1996.

    Google Scholar 

  4. G. Bennett: Inequalities complementary to Hardy. Q. J. Math., Oxf. II. Ser. 49 (1998), 395–432.

    Article  MATH  Google Scholar 

  5. G. Bennett: Series of positive terms. Conf. Proc. Poznań, Poland, 2003 (Z. Ciesielski et al., eds.). Banach Center Publications 64, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2004, pp. 29–38.

    Google Scholar 

  6. G. Bennett: Sums of powers and the meaning of l p. Houston J. Math. 32 (2006), 801–831.

    MATH  MathSciNet  Google Scholar 

  7. F. P. Cass, W. Kratz: Nörlund and weighted mean matrices as operators on l p. Rocky Mt. J. Math. 20 (1990), 59–74.

    Article  MATH  MathSciNet  Google Scholar 

  8. T. Ernst: A Comprehensive Treatment of q-calculus. Birkhäuser, Basel, 2012.

    Book  Google Scholar 

  9. T. Ernst: The History of q-Calculus and a New Method. Uppsala University, Uppsala, 2000, http://www2.math.uu.se/research/pub/Ernst4.pdf.

    Google Scholar 

  10. H. Exton: q-Hypergeometric Functions and Applications. Ellis Horwood Series in Mathematics and Its Applications, Halsted Press, Chichester, 1983.

    MATH  Google Scholar 

  11. P. Gao: A note on Hardy-type inequalities. Proc. Am. Math. Soc. 133 (2005), 1977–1984.

    Article  MATH  Google Scholar 

  12. P. Gao: Hardy-type inequalities via auxiliary sequences. J. Math. Anal. Appl. 343 (2008), 48–57.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Gao: On l p norms of weighted mean matrices. Math. Z. 264 (2010), 829–848.

    Article  MATH  MathSciNet  Google Scholar 

  14. P. Gao: On weighted mean matrices whose l p norms are determined on decreasing sequences. Math. Inequal. Appl. 14 (2011), 373–387.

    MATH  MathSciNet  Google Scholar 

  15. H. Gauchman: Integral inequalities in q-calculus. Comput. Math. Appl. 47 (2004), 281–300.

    Article  MATH  MathSciNet  Google Scholar 

  16. G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities. (2nd ed.), Cambridge University Press, Cambridge, 1952.

    MATH  Google Scholar 

  17. F. H. Jackson: On q-definite integrals. Quart. J. 41 (1910), 193–203.

    MATH  Google Scholar 

  18. V. Kac, P. Cheung: Quantum Calculus. Universitext, Springer, New York, 2002.

    Book  MATH  Google Scholar 

  19. V. Krasniqi: Erratum: Several q-integral inequalities. J. Math. Inequal. 5 (2011), 451.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Kufner, L. Maligranda, L. -E. Persson: The Hardy Inequality. About Its History and Some Related Results. Vydavatelský Servis, Plzeň, 2007.

    MATH  Google Scholar 

  21. A. Kufner, L. Maligranda, L. -E. Persson: The prehistory of the Hardy inequality. Am. Math. Mon. 113 (2006), 715–732.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Kufner, L. -E. Persson: Weighted Inequalities of Hardy Type. World Scientific, Singapore, 2003.

    Book  MATH  Google Scholar 

  23. L. Maligranda: Why Hölder’s inequality should be called Rogers’ inequality. Math. Inequal. Appl. 1 (1998), 69–83.

    MATH  MathSciNet  Google Scholar 

  24. Y. Miao, F. Qi: Several q-integral inequalities. J. Math. Inequal. 3 (2009), 115–121.

    Article  MATH  MathSciNet  Google Scholar 

  25. L. -E. Persson, N. Samko: What should have happened if Hardy had discovered this? J. Inequal. Appl. (electronic only) 2012 (2012), Article ID 29, 11 pages.

  26. M. S. Stanković, P. M. Rajković, S. D. Marinković: On q-fractional derivatives of Riemann-Liouville and Caputo type. arXiv: 0909. 0387v1[math. CA], 2 Sept. 2009.

  27. W. T. Sulaiman: New types of q-integral inequalities. Advances in Pure Math. 1 (2011), 77–80.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87, Luleå, Sweden

    Lech Maligranda & Lars-Erik Persson

  2. L. N. Gumilyev Eurasian National University, Munaytpasov st. 5, 010 008, Astana, Kazakhstan

    Ryskul Oinarov

  3. Narvik University College, P. O. Box 385, N-8505, Narvik, Norway

    Lars-Erik Persson

Authors
  1. Lech Maligranda
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  2. Ryskul Oinarov
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  3. Lars-Erik Persson
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Corresponding author

Correspondence to Lech Maligranda.

Additional information

The second author was supported by the Scientific Committee of Ministry of Education and Science of the Republic of Kazakhstan, grant No. 1529/GF, on priority area “Intellectual potential of the country”.

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Maligranda, L., Oinarov, R. & Persson, LE. On Hardy q-inequalities. Czech Math J 64, 659–682 (2014). https://doi.org/10.1007/s10587-014-0125-6

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  • DOI: https://doi.org/10.1007/s10587-014-0125-6

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