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ON WEIGHTED INEQUALITIES WITH GEOMETRIC MEAN OPERATOR

Part of: Inequalities

Published online by Cambridge University Press:  01 December 2008

DAH-CHIN LUOR
DAH-CHIN LUOR*
Affiliation:
Department of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung 84008, Taiwan (email: [email protected])
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Abstract

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We give a characterization of pairs of weights for the validity of weighted inequalities involving certain generalized geometric mean operators generated by some Volterra integral operators, which include the Hardy averaging operator and the Riemann–Liouville integral operators. The estimations of the constants are also discussed. Our results generalize the work done by J. A. Cochran, C.-S. Lee, H. P. Heinig, B. Opic, P. Gurka, and L. Pick.

Keywords

geometric mean operatorintegral operatorweighted inequalityexponential inequality

MSC classification

Secondary: 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 3 , December 2008 , pp. 463 - 475
Copyright
Copyright © 2009 Australian Mathematical Society

References

[1]Andrews, G. E., Askey, R. and Roy, R., Special Functions, Encyclopedia of Mathematics and its Applications, 71 (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[2]Bradley, J. S., ‘Hardy inequalities with mixed norms’, Canad. Math. Bull. 21 (1978), 405–408.CrossRefGoogle Scholar
[3]Čižmešija, A. and Pečarić, J., ‘Some new generalisations of inequalities of Hardy and Levin–Cochran–Lee’, Bull. Austral. Math. Soc. 63 (2001), 105–113.Google Scholar
[4]Cochran, J. A. and Lee, C.-S., ‘Inequalities related to Hardy’s and Heinig’s’, Math. Proc. Cambridge Philos. Soc. 96 (1984), 1–7.CrossRefGoogle Scholar
[5]Gogatishvili, A., Kufner, A., Persson, L.-E. and Wedestig, A., ‘An equivalence theorem for integral conditions related to Hardy’s inequality’, Real Anal. Exchange 2 (2004), 1–14.Google Scholar
[6]Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, 2nd edn (Cambridge University Press, Cambridge, 1952).Google Scholar
[7]Heinig, H. P., ‘Weighted inequalities in Fourier analysis’, in: Nonlinear Analysis, Function Spaces and Applications, Vol. 4, Teubner-Texte Math., 119 (Teubner, Leipzig, 1990), pp. 42–85.CrossRefGoogle Scholar
[8]Heinig, H. P., ‘Exponential inequalities for a class of operators’, Int. J. Math. Math. Sci. 31 (2002), 283–290.CrossRefGoogle Scholar
[9]Heinig, H. P., Kerman, R. and Krbec, M., ‘Weighted exponential inequalities’, Georgian Math. J. 8 (2001), 69–86.CrossRefGoogle Scholar
[10]Jain, P., Persson, L.-E. and Singh, A. P., ‘On geometric mean inequalities with exponential weights’, Soochow J. Math. 30 (2004), 391–400.Google Scholar
[11]Jain, P., Persson, L.-E. and Wedestig, A., ‘Carleman–Knopp type inequalities via Hardy inequalities’, Math. Inequal. Appl. 4 (2001), 343–355.Google Scholar
[12]Jain, P. and Singh, A. P., ‘A characterization for the boundedness of geometric mean operator’, Appl. Math. Lett. 13 (2000), 63–67.CrossRefGoogle Scholar
[13]Jarrah, A. M. and Singh, A. P., ‘A limiting case of Hardy’s inequality’, Indian J. Math. 43 (2001), 21–36.Google Scholar
[14]Kaijser, S., Nikolova, L., Persson, L.-E. and Wedestig, A., ‘Hardy-type inequalities via convexity’, Math. Inequal. Appl. 8 (2005), 403–417.Google Scholar
[15]Love, E. R., ‘Inequalities related to Knopp’s inequality’, J. Math. Anal. Appl. 137 (1989), 173–180.CrossRefGoogle Scholar
[16]Love, E. R., ‘Inequalities related to those of Hardy and of Cochran and Lee’, Math. Proc. Cambridge Philos. Soc. 99 (1986), 395–408.CrossRefGoogle Scholar
[17]Manakov, V. M., ‘On the best constant in weighted inequalities for Riemann–Liouville integrals’, Bull. London Math. Soc. 24 (1992), 442–448.CrossRefGoogle Scholar
[18]Maz’ja, V. G., Sobolev Spaces (Springer, Berlin, 1985).CrossRefGoogle Scholar
[19]Muckenhoupt, B., ‘Hardy’s inequality with weights’, Studia Math. 44 (1972), 31–38.CrossRefGoogle Scholar
[20]Nassyrova, M., Persson, L.-E. and Stepanov, V. D., ‘On weighted inequalities with geometric mean operator generated by the Hardy-type integral transform’, J. Inequal. Pure Appl. Math. 3 (2002), 48.Google Scholar
[21]Okpoti, C. A., Persson, L.-E. and Sinnamon, G., ‘An equivalence theorem for some integral conditions with general measures related to Hardy’s inequality’, J. Math. Anal. Appl. 326 (2007), 398–413.CrossRefGoogle Scholar
[22]Opic, B. and Gurka, P., ‘Weighted inequalities for geometric means’, Proc. Amer. Math. Soc. 120 (1994), 771–779.CrossRefGoogle Scholar
[23]Opic, B. and Kufner, A., Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219 (Longman Scientific & Technical, Harlow, 1990).Google Scholar
[24]Persson, L.-E. and Stepanov, V. D., ‘Weighted integral inequalities with the geometric mean operator’, J. Inequal. Appl. 7 (2002), 727–746.Google Scholar
[25]Persson, L.-E., Stepanov, V. D. and Ushakova, E. P., ‘Equivalence of Hardy-type inequalities with general measures on the cones of non-negative respective non-increasing functions’, Proc. Amer. Math. Soc. 134 (2006), 2363–2372.CrossRefGoogle Scholar
[26]Pick, L. and Opic, B., ‘On the geometric mean operator’, J. Math. Anal. Appl. 183 (1994), 652–662.CrossRefGoogle Scholar
[27]Sinnamon, G. J., ‘Weighted Hardy and Opial-type inequalities’, J. Math. Anal. Appl. 160 (1991), 434–445.CrossRefGoogle Scholar
[28]Sinnamon, G., ‘Hardy’s inequality and monotonicity’, in: Function Spaces, Differential Operators and Nonlinear Analysis (Mathematical Institute of the Academy of Sciences of the Czech Republic, Prague, 2005), pp. 292–310.Google Scholar
[29]Sinnamon, G. and Stepanov, V. D., ‘The weighted Hardy inequality: new proofs and the case p=1’, J. London Math. Soc. 54 (1996), 89–101.CrossRefGoogle Scholar