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The Scientific World Journal
Volume 2014, Article ID 590656, 15 pages
http://dx.doi.org/10.1155/2014/590656
Research Article

Definition and Properties of the Libera Operator on Mixed Norm Spaces

Faculty of Mathematics, University of Belgrade, Studentski Trg 16, P.O. Box 550, 11001 Beograd, Serbia

Received 7 August 2013; Accepted 23 October 2013; Published 20 February 2014

Academic Editors: H. Jafari and Y. Wang

Copyright © 2014 Miroslav Pavlovic. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. J. Libera, “Some classes of regular uni alent functions,” Proceedings of the American Mathematical Society, vol. 16, pp. 755–758, 1965. View at Google Scholar
  2. A. G. Siskakis, “Composition semigroups and the Ces aro operator on Hp,” Journal of the London Mathematical Society, vol. 36, no. 2, pp. 153–164, 1987. View at Google Scholar
  3. A. G. Siskasis, “Semigroups of composition operators in Bergman spaces,” Bulletin of the Australian Mathematical Society, vol. 35, pp. 397–406, 1987. View at Google Scholar
  4. N. Danikas, S. Ruscheweyh, and A. G. Siskakis, “Metrical and topological properties of a generalized Libera transform,” Archiv der Mathematik, vol. 63, no. 6, pp. 517–524, 1994. View at Publisher · View at Google Scholar · View at Scopus
  5. S. Ruscheweyh and A. G. Siskakis, “Corrigendum to: metrical and topological properties of a generalized Libera transform (Archiv der Mathematik (1994) 63 (517-524)),” Archiv der Mathematik, vol. 91, no. 3, pp. 254–255, 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. M. Nowak and M. Pavlović, “On the Libera operator,” Journal of Mathematical Analysis and Applications, vol. 370, no. 2, pp. 588–599, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. K. Avetisyan and S. Stević, “The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 304–311, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. R. Aulaskari and R. Zhao, “Boundedness and compactness properties of the Libera transform,” in Complex Analysis and Differential Equations (Uppsala, 1997), vol. 64 of Acta Universitatis Upsaliensis. Skrifter rörande Uppsala Universitet. C. Organisation och Historia, pp. 69–80, Uppsala University, Uppsala, Sweden, 1999. View at Google Scholar
  9. D. H. Luecking and L. A. Rubel, Complex Analysis, a Functional Analysis Approach, Springer, New York, NY, USA, 1984.
  10. J. Xiao, “Cesàro-type operators on Hardy, BMOA and Bloch spaces,” Archiv der Mathematik, vol. 68, no. 5, pp. 398–406, 1997. View at Google Scholar · View at Scopus
  11. M. Nowak, “Another proof of boundedness of the Ces aro operator on Hp,” Annales Universitatis Mariae Curie-Skłodowska A, vol. 54, pp. 75–78, 2000. View at Google Scholar
  12. O. Blasco and M. Pavlović, “Coeffcient multipliers on Banach spaces of analytic functions,” Revista Matemática Iberoamericana, no. 2, pp. 415–447, 2011. View at Google Scholar
  13. M. Pavlović, “Analytic functions with decresing coe cients and Hardy spaces and Bloch spaces,” Proceedings of the Edinburgh Mathematical Society, vol. 56, pp. 623–635, 2013. View at Google Scholar
  14. A. Zygmund, Trigonometric Series, vol. 1-2 of Trigonometric, Cambridge University Press, New York, NY, USA, 2nd edition, 1959.
  15. D. Girela, M. Pavlović, and J. Á. Peláez, “Spaces of analytic functions of Hardy-Bloch type,” Journal d'Analyse Mathematique, vol. 100, pp. 53–81, 2006. View at Publisher · View at Google Scholar · View at Scopus
  16. M. Pavlović, “On the moduli of continuity of Hp functions with 0 < p < 1,” Proceedings of the Edinburgh Mathematical Society, vol. 35, pp. 89–100, 1992. View at Google Scholar
  17. A. Zygmund, “Smooth functions,” Duke Mathematical Journal, vol. 12, pp. 47–76, 1945. View at Google Scholar
  18. M. Pavlović, Introduction to Function Spaces on the Disk, Matemati cki Institut u Beogradu, Posebna Izdanja, 20th edition, 2004.
  19. S. Stević, “On Libera-type transforms on the unit disc, polydisc and the unit ball,” Integral Transforms and Special Functions, vol. 19, no. 11, pp. 785–799, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. J. I.-H. Shi and G.-B. Ren, “Boundedness of the Cesàro operator on mixed norm spaces,” Proceedings of the American Mathematical Society, vol. 126, no. 12, pp. 3553–3560, 1998. View at Google Scholar · View at Scopus
  21. Z. Hu and T. Liu, “Weighted integrals of holomorphic functions,” Applied Mathematics, vol. 19, no. 4, pp. 474–480, 2004. View at Google Scholar
  22. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.
  23. M. Mateljević and M. Pavlović, “Lp-behavior of power series with positive coeffcients and Hardy spaces,” Proceedings of the American Mathematical Society, vol. 87, no. 2, pp. 309–316, 1983. View at Google Scholar
  24. M. Jevtić and M. Pavlović, “On multipliers from Hp to q (0 <  q < p < 1),” Archiv der Mathematik, vol. 56, pp. 174–180, 1991. View at Google Scholar
  25. M. Jevtić and M. Pavlović , “Coeffcient multipliers on spaces of analytic functions,” Acta Scientiarum Mathematicarum, vol. 64, pp. 531–545, 1998. View at Google Scholar
  26. G. H. Hardy and J. E. Littlewood, “Some properties of fractional integrals. II,” Mathematische Zeitschrift, vol. 34, no. 1, pp. 403–439, 1932. View at Publisher · View at Google Scholar · View at Scopus
  27. T. M. Flett, “Lipschitz spaces of functions on the circle and the disc,” Journal of Mathematical Analysis and Applications, vol. 39, no. 1, pp. 125–158, 1972. View at Google Scholar · View at Scopus
  28. J. Khintchin and A. N. Kolmogorov, “Uber Kon ergenz on Reihen deren Glieder durch den Zufall bestimmt werden,” Matematicheskii Sbornik, vol. 32, pp. 123–138, 1925. View at Google Scholar
  29. M. Mateljević and M. Pavlović, “Lp-behaviour of the integral means of analytic functions,” Studia Mathematica, vol. 77, pp. 219–237, 1984. View at Google Scholar
  30. S. V. Kisliakov, “Fourier coeffcients of boundary values of analytic functions on the disc and the bidisc,” Trudy Matematicheskogo Instituta Imeni V. A. Steklova, vol. 155, pp. 77–91, 1981. View at Google Scholar