Korean J. Math. Vol. 22 No. 3 (2014) pp.407-417
DOI: https://doi.org/10.11568/kjm.2014.22.3.407

Some proofs of the classical integral Hardy inequality

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Published: Sep 30, 2014
Keywords:
Hardy inequality, Proofs, homogeneity, integral oper
Mathematics Subject Classification:
26D10, 26D15

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Mohammed Muniru Iddrisu
University for Development Studies, Faculty of Mathematical Sciences, Department of Mathematics, Navrongo campus, Ghana
Adjei Christopher Okpoti
Department of Mathematics University of Education Winneba, Ghana
Alagbe Kazeem Gbolagade
Department of Computer Science University for Development Studies Navrongo Campus P. O. Box 24, Navrongo, UE/R, Ghana

Abstract

We present some proofs of the classical integral Hardy inequality. Our approach makes use of continuous functions with compact support in $(0,\infty)$, homogeneity of the norm and Schur's criterion for integral operators.



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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. Google Scholar

[2] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., John Willey and Sons, Inc., New York, 1999. Google Scholar

[3] C. Gasquet and P. Witomski, Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets, Springer-Verlag New York, Inc., 1999. Google Scholar

[4] G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), 314–317. Google Scholar

[5] G. H. Hardy, J. E. Littlewood and G.Polya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952. Google Scholar

[6] S. Kaijser, L. Nikolova, L.- E. Persson and A. Wedestig, Hardy-Type Inequalities Via Convexity, J. Math. Ineq. Appl., 8 (3) (2005), 403–417. Google Scholar

[7] A. Kufner, L. Maligranda and L.- E. Persson, The Hardy inequality: About its History and Some Related Results, Vydavatelsky Sevis Publishing House, Pilsen, 2007. Google Scholar

[8] A. Kufner, L. Maligranda and L.-E. Persson, The Prehistory of the Hardy Inequality, Amer. Math. Monthly, 113 (8) (2006), 715–732. Google Scholar

[9] J. A. Oguntuase and L.- E. Persson, Refinement of Hardy inequalities via superquadratic and subquadratic functions, J. Math. Anal. Appl., 339 (2) (2008), 1305–1312. Google Scholar

[10] W. Rudin, Real and Complex Analysis, 2nd edition, McGraw-Hill Inc, 1974. Google Scholar

[11] T. Tao, Analysis I, Hindustan Book Agency, Volume I, 2006. Google Scholar

[12] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, NewYork, 2005. Google Scholar