On Bloch-Type Functions with Hadamard Gaps

Stevic, Stevo

doi:https://doi.org/10.1155/2007/39176

Abstract and Applied Analysis

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Volume 2007 | Article ID 039176 | https://doi.org/10.1155/2007/39176

On Bloch-Type Functions with Hadamard Gaps

Stevo Stevic¹

Academic Editor: Simeon Reich

Received02 May 2007

Accepted20 Aug 2007

Published11 Nov 2007

Abstract

We give some sufficient and necessary conditions for an analytic function f on the unit ball B with Hadamard gaps, that is, for f(z)=∑k=1∞Pnk(z) (the homogeneous polynomial expansion of f) satisfying nk+1/nk≥λ>1 for all k∈ℕ, to belong to the space ℬpα(B)={f|sup0<r<1(1−r2)α\|ℛfr\|p<∞,f∈H(B)}, p=1,2,∞ as well as to the corresponding little space. A remark on analytic functions with Hadamard gaps on mixed norm space on the unit disk is also given.

References

D. D. Clahane and S. Stević, “Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,” Journal of Inequalities and Applications, vol. 2006, Article ID 61018, p. 11, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
S. Li, “Fractional derivatives of Bloch-type functions,” Siberian Mathematical Journal, vol. 46, no. 2, pp. 394–402, 2005.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Nowak, “Bloch space on the unit ball of $ℂ^{n}$ ,” Annales Academiæ Scientiarium Fennicæ, vol. 23, no. 2, pp. 461–473, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Nowak, “Bloch space and Möbius invariant Besov spaces on the unit ball of $ℂ^{n}$ ,” Complex Variables, Theory and Application, vol. 44, no. 1, pp. 1–12, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. H. Ouyang, W. S. Yang, and R. H. Zhao, “Characterizations of Bergman spaces and Bloch space in the unit ball of $ℂ^{n}$ ,” Transactions of the American Mathematical Society, vol. 347, no. 11, pp. 4301–4313, 1995.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. M. Timoney, “Bloch functions in several complex variables. I,” The Bulletin of the London Mathematical Society, vol. 12, no. 4, pp. 241–267, 1980.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Yamashita, “Gap series and $α$ -Bloch functions,” Yokohama Mathematical Journal, vol. 28, no. 1-2, pp. 31–36, 1980.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. S. Choa, “Some properties of analytic functions on the unit ball with Hadamard gaps,” Complex Variables, Theory and Application, vol. 29, no. 3, pp. 277–285, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Miao, “A property of analytic functions with Hadamard gaps,” Bulletin of the Australian Mathematical Society, vol. 45, no. 1, pp. 105–112, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Mateljević and M. Pavlović, “ $L^{p}$ -behavior of power series with positive coefficients and Hardy spaces,” Proceedings of the American Mathematical Society, vol. 87, no. 2, pp. 309–316, 1983.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Zygmund, Trigonometric Series. Vol. I, Cambridge University Press, New York, NY, USA, 2nd edition, 1959.
View at: Zentralblatt MATH | MathSciNet
S. Stević, “A generalization of a result of Choa on analytic functions with Hadamard gaps,” Journal of the Korean Mathematical Society, vol. 43, no. 3, pp. 579–591, 2006.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Girela and J. A. Peláez, “Integral means of analytic functions,” Ann. Acad. Sci. Fenn., vol. 29, pp. 459–469, 2004.
View at: Google Scholar

Copyright

Copyright © 2007 Stevo Stević. 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.

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