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Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales

  • S. H. Saker EMAIL logo and Donal O’Regan
Published/Copyright: December 9, 2015
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 65 Issue 5

Abstract

In this paper using some algebraic inequalities, Hölder inequality and a simple consequence of Keller’s chain rule we prove some new inequalities of Hardy type on a time scale T. These inequalities as special cases contain some integral and discrete inequalities when T = ℝ and T = ℕ.

Keywords: Hardy’s inequality; time scales

References

[1] ACHPATTE, B. G.: A note on some extensions of Hardy’s inequality, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) XLIV (1998), 95-100.Search in Google Scholar

[2] BESSACK, P. R.: Hardy’s inequality and its extensions, Pacific J. Math. 11 (1961), 39-61.10.2140/pjm.1961.11.39Search in Google Scholar

[3] BOHNER, M.-PETERESON, A.: Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, MA, 2001.10.1007/978-1-4612-0201-1Search in Google Scholar

[4] BOHNER, M.-PETERSON, A.: Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, MA, 2003.10.1007/978-0-8176-8230-9Search in Google Scholar

[5] CHAN, L. Y.: Some extensions of Hardy’s inequality, Canad. Math. Bull. 22 (1979), 165-169.10.4153/CMB-1979-023-5Search in Google Scholar

[6] CHANG-JIAN, Z.-LIAN-YING, C.-CHEUNG, W-S.: On some new Hilbert-type inequalities, Math. Slovaca 61 (2012), 15-28.10.2478/s12175-010-0056-0Search in Google Scholar

[7] HARDY, G. H.: Notes on a theorem of Hilbert, Math. Z. 6 (1920), 314-317.10.1007/BF01199965Search in Google Scholar

[8] HARDY, G. H.: Notes on some points in the integral calculus, Messenger Math. 57 (1928), 12-16.Search in Google Scholar

[9] HARDY, G. H.-LITTLEWOOD, J. E.-POLYA, G.: Inequalities (2nd ed.), Cambridge Univ. Press, Cambridge, 1952.Search in Google Scholar

[10] HILGER, S.: Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18-56.10.1007/BF03323153Search in Google Scholar

[11] IZUMI, M.-IZUMI, S.: On some inequlities for Fourier Series, J.Anal.Math. 21 (1968), 277-291.10.1007/BF02787671Search in Google Scholar

[12] KAAIJSER, S.-NIKOLOVA, L.-PERSOON, L. E.-WEDESTIG, A.: Hardy-type inequalities via convexity, Math. Inequal. Appl. 8 (2005), 403-417.Search in Google Scholar

[13] KAAIJSER, S.-PERSOON, L. E.-¨OBERG, A.: On Carleman and Knopp’s inequalities, J. Approx. Theory 117 (2002), 140-151.10.1006/jath.2002.3684Search in Google Scholar

[14] KUFNER, K.-MALIGRANDA, L.-PERSOON, L. E.: The Hardy inequalities: About its history and some related results, Pilsen (2007).Search in Google Scholar

[15] LEVINSON, N.: Generalizations of an inequality of Hardy, Duke Math. J. 31 (1964), 389-394.Search in Google Scholar

[16] OGUNTUASE, J. A.-ADELEKE, E. O.: On Hardy’s integral inequality, Facta Univ. Ser. Math. Inform. 20 (2005), 9-20.Search in Google Scholar

[17] OPIC, B.-KUFNER, K.: Hardy-type Inequalities, Longman Scientific& Technical, Harlow-Essex, 1989.Search in Google Scholar

[18] OZKAN, U. M.-YILDIRM, H.: Hardy-Knopp-type inequalities on time scales, Dynam. Systems Appl. 17 (2008), 477-486.Search in Google Scholar

[19] PACHPATTE, B. G.: A note on certain inequalities related to Hardy’s inequality, Indian J. Pure Appl. Math 23 (1992), 773-776.Search in Google Scholar

[20] PAN, J.: Geometric character of domain and the Harnack inequality of solution of a singular parabolic equation, Math. Slovaca 62 (2012), 721-734.10.2478/s12175-012-0040-ySearch in Google Scholar

[21] ŘEHAK, P.: Hardy inequality on time scales and its application to half-linear dynamic equations, J. Inequal. Appl. 2005 (2005), 495-507.10.1155/JIA.2005.495Search in Google Scholar

[22] SIDI, M. R.-Torres, A. F. M.: H¨older’s and Hardy’s two dimensional diamond-alpha inequalities on time scales, Ann. Univ. Craiva, Math. Comp. Ser. 37 (2010), 1-11.Search in Google Scholar

[23] SPEDDING, V.: Taming Nature’s Numbers, New Scientist 2404 (2003), 28-31. Search in Google Scholar

[24] TUNA, A.-KUTUKCU, S.: Some integrals inequalities on time scales, Appl. Math. Mech. (English Ed.) 29 (2008), 23-29.10.1007/s10483-008-0104-ySearch in Google Scholar

Received: 2012-11-13
Accepted: 2013-4-1
Published Online: 2015-12-9
Published in Print: 2015-10-1

Mathematical Institute Slovak Academy of Sciences

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  1. Finite Mixed Sums wih Harmonic Terms
  2. Packing of ℝ2 by Crosses
  3. On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
  4. Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
  5. Generalized Derivations on Lie Ideals and Power Values on Prime Rings
  6. On Monoids of Injective Partial Cofinite Selfmaps
  7. Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
  8. The Controlled Convergence Theorem for the Gap-Integral
  9. The Solvability of a Nonlocal Boundary Value Problem
  10. Oscillation Criteria for Third Order Differential Equations with Functional Arguments
  11. Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
  12. On Null Lagrangians
  13. Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
  14. Convergence of Series on Large Set of Indices
  15. On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
  16. Representation of Extendible Bilinear Forms
  17. Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
  18. Topological Fundamental Groups and Small Generated Coverings
  19. A Relation between two Kinds of Norms for Martingales
  20. Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
  21. Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
  22. Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
https://doi.org/10.1515/ms-2015-0068
Keywords for this article
Hardy’s inequality; time scales

Articles in the same Issue

  1. Finite Mixed Sums wih Harmonic Terms
  2. Packing of ℝ2 by Crosses
  3. On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
  4. Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
  5. Generalized Derivations on Lie Ideals and Power Values on Prime Rings
  6. On Monoids of Injective Partial Cofinite Selfmaps
  7. Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
  8. The Controlled Convergence Theorem for the Gap-Integral
  9. The Solvability of a Nonlocal Boundary Value Problem
  10. Oscillation Criteria for Third Order Differential Equations with Functional Arguments
  11. Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
  12. On Null Lagrangians
  13. Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
  14. Convergence of Series on Large Set of Indices
  15. On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
  16. Representation of Extendible Bilinear Forms
  17. Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
  18. Topological Fundamental Groups and Small Generated Coverings
  19. A Relation between two Kinds of Norms for Martingales
  20. Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
  21. Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
  22. Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
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