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  • Research Article
  • Open Access

Boundary behaviour of analytic functions in spaces of Dirichlet type

Journal of Inequalities and Applications20062006:927957

  • Received: 24 June 2005
  • Accepted: 8 November 2005
  • Published:


For and , we let be the space of all analytic functions in such that belongs to the weighted Bergman space . We obtain a number of sharp results concerning the existence of tangential limits for functions in the spaces . We also study the size of the exceptional set , where denotes the radial variation of along the radius , for functions .


  • Analytic Function
  • Bergman Space
  • Radial Variation
  • Boundary Behaviour
  • Weighted Bergman Space


Authors’ Affiliations

Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, Málaga, 29071, Spain
Departamento de Matemática Aplicada, Escuela Politécnica, Universidad de Málaga, Campus de El Ejido, Málaga, 29071, Spain


  1. Ahern PR: The mean modulus and the derivative of an inner function. Indiana University Mathematics Journal 1979,28(2):311–347. 10.1512/iumj.1979.28.28022MathSciNetView ArticleMATHGoogle Scholar
  2. Ahern PR, Clark DN: On inner functions with -derivative. The Michigan Mathematical Journal 1974, 21: 115–127.MathSciNetView ArticleMATHGoogle Scholar
  3. Arazy J, Fisher SD, Peetre J: Möbius invariant function spaces. Journal für die reine und angewandte Mathematik 1985, 363: 110–145.MathSciNetMATHGoogle Scholar
  4. Baernstein A II, Girela D, Peláez JÁ: Univalent functions, Hardy spaces and spaces of Dirichlet type. Illinois Journal of Mathematics 2004,48(3):837–859.MathSciNetMATHGoogle Scholar
  5. Beurling A: Ensembles exceptionnels. Acta Mathematica 1939, 72: 1–13.MathSciNetView ArticleMATHGoogle Scholar
  6. Buckley SM, Koskela P, Vukotić D: Fractional integration, differentiation, and weighted Bergman spaces. Mathematical Proceedings of the Cambridge Philosophical Society 1999,126(2):369–385. 10.1017/S030500419800334XMathSciNetView ArticleMATHGoogle Scholar
  7. Cargo GT: Angular and tangential limits of Blaschke products and their successive derivatives. Canadian Journal of Mathematics 1962, 14: 334–348. 10.4153/CJM-1962-026-2MathSciNetView ArticleMATHGoogle Scholar
  8. Collingwood EF, Lohwater AJ: The Theory of Cluster Sets, Cambridge Tracts in Mathematics and Mathematical Physics, no. 56. Cambridge University Press, Cambridge; 1966:xi+211.View ArticleGoogle Scholar
  9. Donaire JJ, Girela D, Vukotić D: On univalent functions in some Möbius invariant spaces. Journal für die reine und angewandte Mathematik 2002, 553: 43–72.MATHGoogle Scholar
  10. Duren PL: Theory of H p Spaces, Pure and Applied Mathematics. Volume 38. Academic Press, New York; 1970:xii+258. reprint: Dover, New York, 2000Google Scholar
  11. Duren PL, Schuster AP: Bergman Spaces, Mathematical Surveys and Monographs. Volume 100. American Mathematical Society, Rhode Island; 2004:x+318.MATHGoogle Scholar
  12. Flett TM: The dual of an inequality of Hardy and Littlewood and some related inequalities. Journal of Mathematical Analysis and Applications 1972, 38: 746–765. 10.1016/0022-247X(72)90081-9MathSciNetView ArticleMATHGoogle Scholar
  13. Girela D, Peláez JÁ: Non-stable classes of analytic functions. International Journal of Pure and Applied Mathematics 2005,21(4):553–563.MathSciNetMATHGoogle Scholar
  14. Girela D, Peláez JÁ: Carleson measures for spaces of Dirichlet type. to appear in Integral Equations and Operator Theory to appear in Integral Equations and Operator TheoryGoogle Scholar
  15. Girela D, Peláez JÁ: Growth properties and sequences of zeros of analytic functions in spaces of Dirichlet type. to appear in Journal of the Australian Mathematical Society to appear in Journal of the Australian Mathematical SocietyGoogle Scholar
  16. Hedenmalm H, Korenblum B, Zhu KH: Theory of Bergman Spaces, Graduate Texts in Mathematics. Volume 199. Springer, New York; 2000:x+286.MATHGoogle Scholar
  17. Kahane J-P, Salem R: Ensembles parfaits et séries trigonométriques, Actualités Sci. Indust., no. 1301. Hermann, Paris; 1963:192.Google Scholar
  18. Kim HO: Derivatives of Blaschke products. Pacific Journal of Mathematics 1984,114(1):175–190.MathSciNetView ArticleMATHGoogle Scholar
  19. Kinney JR: Tangential limits of functions of the class . Proceedings of the American Mathematical Society 1963, 14: 68–70.MathSciNetMATHGoogle Scholar
  20. Littlewood JE: On a theorem of Fatou. Journal of the London Mathematical Society 1927, 2: 172–176. 10.1112/jlms/s1-2.3.172MathSciNetView ArticleMATHGoogle Scholar
  21. Littlewood JE, Paley REAC: Theorems on Fourier series and power series. II. Proceedings of the London Mathematical Society 1936, 42: 52–89.MathSciNetMATHGoogle Scholar
  22. Lohwater AJ, Piranian G: The boundary behavior of functions analytic in a disk. Annales Academiae Scientiarum Fennicae. Series A I 1957,1957(239):17.MathSciNetMATHGoogle Scholar
  23. Nagel A, Rudin W, Shapiro JH: Tangential boundary behavior of functions in Dirichlet-type spaces. Annals of Mathematics. Second Series 1982,116(2):331–360. 10.2307/2007064MathSciNetView ArticleMATHGoogle Scholar
  24. Protas D: Blaschke products with derivative in and . The Michigan Mathematical Journal 1973, 20: 393–396.MathSciNetMATHGoogle Scholar
  25. Rudin W: The radial variation of analytic functions. Duke Mathematical Journal 1955,22(2):235–242. 10.1215/S0012-7094-55-02224-9MathSciNetView ArticleMATHGoogle Scholar
  26. Twomey JB: Tangential limits for certain classes of analytic functions. Mathematika 1989,36(1):39–49. 10.1112/S0025579300013553MathSciNetView ArticleMATHGoogle Scholar
  27. Twomey JB: Radial variation of functions in Dirichlet-type spaces. Mathematika 1997,44(2):267–277. 10.1112/S0025579300012596MathSciNetView ArticleMATHGoogle Scholar
  28. Vinogradov SA: Multiplication and division in the space of analytic functions with an area-integrable derivative, and in some related spaces. Rossiĭ skaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheskiĭ Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI) 1995,222(23):45–77, 308. translation in Journal of Mathematical Sciences (New York) 87 (1997), no. 5, 3806–3827, Issled. po Linein. Oper. i Teor. Funktsii translation in Journal of Mathematical Sciences (New York) 87 (1997), no. 5, 3806–3827, Issled. po Linein. Oper. i Teor. FunktsiiMATHGoogle Scholar
  29. Zhu KH: Analytic Besov spaces. Journal of Mathematical Analysis and Applications 1991,157(2):318–336. 10.1016/0022-247X(91)90091-DMathSciNetView ArticleMATHGoogle Scholar
  30. Zygmund A: On certain integrals. Transactions of the American Mathematical Society 1944,55(2):170–204. 10.2307/1990189MathSciNetView ArticleMATHGoogle Scholar
  31. Zygmund A: On a theorem of Littlewood. Summa Brasiliensis Mathematicae 1949,2(5):51–57.MathSciNetMATHGoogle Scholar
  32. Zygmund A: Trigonometric Series. Vols. I, II. 2nd edition. Cambridge University Press, New York; 1959:xii+383, vii+354.Google Scholar


© D. Girela and J.Á. Peláez 2006

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