Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On a theorem of Privalov and normal functions


Author: Daniel Girela
Journal: Proc. Amer. Math. Soc. 125 (1997), 433-442
MSC (1991): Primary 30D45, 30D55
DOI: https://doi.org/10.1090/S0002-9939-97-03544-2
MathSciNet review: 1363422
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A well known result of Privalov asserts that if $f$ is a function which is analytic in the unit disc $\Delta =\{z\in \mathbb {C} : \vert z\vert <1\} $, then $f$ has a continuous extension to the closed unit disc and its boundary function $f(e\sp {i\theta })$ is absolutely continuous if and only if $f\sp {\prime }$ belongs to the Hardy space $H\sp {1}$. In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, $M_{1}(r, f\sp {\prime })= \frac {1}{2\pi }\int _{-\pi }\sp {\pi }\left \vert f\sp {\prime }(re\sp {i\theta }) \right \vert \, d\theta ,$ we prove that for any positive continuous function $\phi $ defined in $(0, 1)$ with $\phi (r)\to \infty $, as $r\to 1$, there exists a function $f$ analytic in $\Delta $ which is not a normal function and with the property that $M_{1}(r, f\sp {\prime })\leq \phi (r) $, for all $r$ sufficiently close to $1$.


References [Enhancements On Off] (What's this?)

  • [1] J. M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12-37. MR 50:13536
  • [2] A. Baernstein and J. E. Brown, Integral means of derivatives of monotone slit mappings, Comment. Math. Helvetici 57 (1982), 331-348. MR 85j:30033
  • [3] C. Bennett and M. Stoll, Derivatives of analytic functions and bounded mean oscillation, Arch. Math. 47 (1986), 438-442. MR 88a:30074
  • [4] R. D. Berman, L. Brown and W. S. Cohn, Moduli of continuity and generalized BCH sets, Rocky Mountain J. Math. 17 (3) (1987), 315-338. MR 88j:30010
  • [5] O. Blasco and G. Soares de Souza, Spaces of analytic functions on the disc where the growth of $M_{p}(F, r)$ depends on a weight, J. Math. Anal. Appl. 147 (2) (1990), 580-598. MR 92e:46051
  • [6] P. Bourdon, J. Shapiro and W. Sledd, Fourier series, mean Lipschitz spaces and bounded mean oscillation, Analysis at Urbana 1, Proceedings of the Special Year in Modern Analysis at the University of Illinois, 1986-87 (E. R. Berkson, N. T. Peck and J. Uhl, eds.), London Math. Soc. Lecture note series 137, Cambridge Univ. Press, 1989, pp. 81-110. MR 90j:42011
  • [7] J. A. Cima and K. E. Petersen, Some analytic functions whose boundary values have bounded mean oscillation, Math. Z. 147 (1976), 237-247. MR 53:8431
  • [8] P. L. Duren, Theory of $H\sp {p}$ spaces, Academic Press (New York), 1970. MR 42:3552
  • [9] O. Lehto and K. J. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47-65. MR 19:403
  • [10] J. Lippus, On multipliers preserving the classes of functions with a given major of the modulus of continuity, J. Approx. Th. 66 (1991), 190-197. MR 92m:42010
  • [11] K. I. Oskolkov, An estimate of the rate of approximation of a continuous function and its conjugate by Fourier sums on a set of total measure, Izv. Acad. Nauk SSSR Ser. Mat. 38 (1974), no.6, 1393-1407 (Russian); English transl. in Math. USSR Izv. 8 (6) (1974), no. 6, 1372-1386 (1976). MR 50:10663
  • [12] K. I. Oskolkov, Uniform modulus of continuity of summable functions on sets of positive measure, Dokl. Akad. Nauk SSSR 229 (1976), no.2, 304-306 (Russian); English transl. in Sov. Math. Dokl. 17 (1976), no. 4, 1028-1030 (1977). MR 57:9917
  • [13] K. I. Oskolkov, Approximation properties of summable functions on sets of full measure, Mat. Sbornik, n. Ser. 103(145) (1977), 563-589 (Russian); English transl. in Math. USSR Sbornik 32 (1977), no. 4, 489-514 (1978). MR 57:13343
  • [14] K. I. Oskolkov, On Luzin's C-property for a conjugate function, Trudy Mat. Inst. Steklova 164 (1983), 124-135 (Russian); English transl. in Proc. Steklov Inst. Math. 164 (1985), 141-153. MR 86e:42019
  • [15] Ch. Pommerenke, Univalent Functions, Vandenhoeck und Ruprecht, Göttingen, 1975. MR 58:22526
  • [16] D. Protas, Blaschke products with derivative in $H\sp {p}$ and $B\sp {p}$, Michigan Math. J. 20 (1973), 393-396. MR 49:9217
  • [17] S. Yamashita, A non-normal function whose derivative has finite area integral of order $0<p<2$, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (2) (1979), 293-398. MR 81k:34002
  • [18] S. Yamashita, A non-normal function whose derivative is of Hardy class $H\sp {p}$, $0<p<1$, Canad. Math. Bull. 23 (4) (1980), 499-500. MR 82a:34006

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30D45, 30D55

Retrieve articles in all journals with MSC (1991): 30D45, 30D55


Additional Information

Daniel Girela
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
Email: Girela@ccuma.sci.uma.es

DOI: https://doi.org/10.1090/S0002-9939-97-03544-2
Keywords: Normal functions, Hardy spaces, integral means, theorem of Privalov
Received by editor(s): November 1, 1994
Received by editor(s) in revised form: June 25, 1995
Additional Notes: This research has been supported in part by a D.G.I.C.Y.T. grant (PB91-0413) and by a grant from “La Junta de Andalucía”
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society