Illinois Journal of Mathematics

Argument of bounded analytic functions and Frostman’s type conditions

Igor Chyzhykov

Full-text: Open access

Abstract

We describe the growth of the naturally defined argument of a bounded analytic function in the unit disk in terms of the complete measure introduced by A. Grishin. As a consequence, we characterize the local behavior of a logarithm of an analytic function. We also find necessary and sufficient conditions for closeness of $\log f(z)$, $f\in H^\infty$, and the local concentration of the zeros of $f$.

Article information

Source
Illinois J. Math., Volume 53, Number 2 (2009), 515-531.

Dates
First available in Project Euclid: 23 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1266934790

Digital Object Identifier
doi:10.1215/ijm/1266934790

Mathematical Reviews number (MathSciNet)
MR2594641

Zentralblatt MATH identifier
1188.30063

Subjects
Primary: 30D50

Citation

Chyzhykov, Igor. Argument of bounded analytic functions and Frostman’s type conditions. Illinois J. Math. 53 (2009), no. 2, 515--531. doi:10.1215/ijm/1266934790. https://projecteuclid.org/euclid.ijm/1266934790


Export citation

References

  • P. R. Ahern and D. N. Clark, Radial $N^{th}$ derivatives of Blaschke products, Math. Scan. 28 (1971), 189–201.
  • G. T. Cargo, Angular and tangential limits of Blaschke products and their successive derivatives, Canad. J. Math. 14 (1962), 334–348.
  • I. Chyzhykov, A generalization of Hardy–Littlewood's theorem, Math. Methods and Physicomechanical Fields 49 (2006), 74–79 (in Ukrainain).
  • I. Chyzhykov, Growth and representation of analytic and harmonic functions in the unit disk, Ukr. Math. Bull. 3 (2006), 31–44.
  • W. S. Cohn, Radial limits and star invariant subspaces of bounded mean oscillation, Amer. J. Math. 108 (1986), 719–749.
  • E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Univ. Press, 1966.
  • M. M. Djrbashian, Integral transforms and representations of functions in the complex domain, Nauka, Moscow, 1966 (in Russian).
  • M. A. Fedorov and A. F. Grishin, Some questions of the Nevanlinna theory for the complex half-plane, Math. Physics, Analysis and Geometry (Kluwer Acad. Publish.) 1 (1998), 223–271.
  • O. Frostman, Sur le produits de Blaschke, K. Fysiogr. Sallsk. Lund Forh. 12 (1939), 1–14.
  • D. Girela, J. A. Peláez and D. Vukotić, Intergrability of the derivative of a Blaschke product, Proc. Edinburg Math. Soc. 50 (2007), 673–688.
  • A. Grishin, Continuity and asymptotic continuity of subharmonic functions, Math. Physics, Analysis, Geometry, ILPTE 1 (1994), 193–215 (in Russian).
  • G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Zeitschrift 34 (1932), 403–439.
  • W. K. Hayman, Meromorphic functions, Oxford, Clarendon Press, 1964.
  • W. K. Hayman and P. B. Kennedy, Subharmonic functions, vol. 1, Academic Press, London, 1976.
  • W. K. Hayman, The minimum modulus of large integral functions, Proc. London Math. Soc. (3) 2 (1952), 469–512.
  • I. F. Krasichkov, Lower bounds for entire functions of finite order, Siberian Math. J. 6 (1965), 840–861 (in Russian).
  • C. N. Linden, The minimum modulus of functions regular and of finite order in the unit circle, Quart. J. Math. 7 (1956), 196–216.
  • Ya. V. Mykytyuk and Ya. V. Vasylkiv, Criteria for boundedness of the integral means of logarithms of Blaschke products, Dopov. Nats. Akad. Nauk Ukr., Mat. Prirodozn Tekh. Nauki 8 (2000), 10–14 (in Ukrainian).
  • A. V. Rybkin, Convergence of arguments of Blaschke products in $L_p$-metrics, Proc. Amer. Math. Soc. 111 (1991), 701–708.
  • M. Tsuji, Canonical product for a meromorphic function in a unit circle, J. Math. Soc. Japan 8 (1956), 7–21.
  • A. Zygmund, Trigonometric series, vol. 1, 2, Cambridge Univ. Press, 1959.