Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

APPLICATIONS OF TAYLOR SERIES FOR CARLEMAN'S INEQUALITY THROUGH HARDY INEQUALITY

  • Received : 2015.08.24
  • Accepted : 2015.10.15
  • Published : 2015.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we prove the discrete Hardy inequality through the continuous case for decreasing functions using elementary properties of calculus. Also, we prove the Carleman's inequality through limiting the discrete Hardy inequality with applications of Taylor series.

Keywords

References

  1. S. Abramovich, K. Krulic, J. Pecaric and L.E. Persson, Some new refined Hardy type inequalities with general kernels and measures Aequat. Math. 79 (2010), 157-172. https://doi.org/10.1007/s00010-010-0011-8
  2. A. Cizmesija, J. Pecaric and L.E. Persson, On Strengthened Hardy and Polya-Knopp inequalities J. Approx. Theory 125 (2003), 74-84. https://doi.org/10.1016/j.jat.2003.09.007
  3. G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, (1952)
  4. G.H. Hardy, Note on a theorem of Hilbert, Math. Z. 6, (1920), 314-317. https://doi.org/10.1007/BF01199965
  5. M. Johansson, L.E. Persson and A. Wedestig, Carleman's inequality- History, Proofs and Some new generalizations, J. Inequal. Pure Appl. Math. 4 (2003), 1443-5756.
  6. M. Johansson, Carleman type inequalities and Hardy type inequalities for monotone functions, Ph.D thesis, Lulea University of Technology, Sweden, (2007)
  7. S. Kaijser, L. Nikolova, L.E. Persson and A. Wedestig, Hardy-Type Inequalities Via Convexity, J. Math. Ineq. Appl. 8 (2005), 403-417.
  8. S. Kaijser, L. Nikolova, L.E. Persson and A. Oberg, On Carleman and Knopp's Inequalities, J. Approx. Theory 117 (2002), 140-151. https://doi.org/10.1006/jath.2002.3684
  9. A. Kufner, L. Maligranda and L.E. Persson, The Prehistory of the Hardy Inequality, Amer. Math. Monthly 113 (2006), 715-732. https://doi.org/10.2307/27642033
  10. A. Kufner, L. Maligranda and L.E. Persson, The Hardy inequality: About its History and Some Related Results, Vydavatelsky Sevis Publishing House, Pilsen, (2007)
  11. J.A. Oguntuase and L.E. Persson, Refinement of Hardy inequalities via superquadrattic and subquadratic functions, J. Math. Anal. Appl. 339 (2008), 1305-1312. https://doi.org/10.1016/j.jmaa.2007.08.007
  12. J.A. Oguntuase, L.E. Persson, E.K. Essel and B.A. Popoola, Refined multidimensional Hardy-Type Inequalities Via Superquadracity, Banach J. Math. Anal. 2 (2008), 129-139. https://doi.org/10.15352/bjma/1240336299
  13. C.A. Okpoti, L.E. Persson and A. Wedestig, A Scales of Weight Characterizations for discrete Hardy and Carleman type inequalities, Proc. Function Spaces, Differential Operators and Nonlinear Analysis (FS-DONA 2004), Math. Institute, Acad. Sci., Czech Republic, Milovy, (2004), 236-258.
  14. C.A. Okpoti, Weight characterizations of Discrete Hardy and Carleman's type inequalities, Ph.D thesis, Lulea University of Technology, Sweden, (2005)
  15. J. Pecaric and K.B. Stolarsky, Carleman's inequality: History and new generalizations, Aequat. Math. 61 (2001), 49-62. https://doi.org/10.1007/s000100050160