Journal d'Analyse Mathématique

, Volume 134, Issue 1, pp 107–126 | Cite as

On the existence of strips inside domains convex in one direction

  • Dimitrios Betsakos


A plane domain Ω is convex in the positive direction if for every ωΩ, the entire half-line {ω + t: t ≥ 0} is contained in Ω. Suppose that h maps the unit disk onto such a domain Ω with the normalization h(0) = 0 and limt→∞h−1(h(z) + t) = 1. We show that if ∠limz→−1 Re h(z) = −∞ and ∠limz→−1(1 + z)h′(z) = ν ∈ (0, +∞), then Ω contains a maximal horizontal strip of width πν. We also prove a converse statement. These results provide a solution to a problem posed by Elin and Shoikhet in connection with semigroups of holomorphic functions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Quasiconformal Mappings and Their Applications, Narosa, New Delhi, 2007, pp. 9–56.zbMATHGoogle Scholar
  2. [2]
    J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy Transform, American Mathematical Society, Providence, RI, 2006.CrossRefzbMATHGoogle Scholar
  3. [3]
    E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets, Cambridge Univ. Press, Cambridge, 1966.CrossRefzbMATHGoogle Scholar
  4. [4]
    M. D. Contreras and S. Diaz-Madrigal, Analytic flows on the unit disk: angular derivatives and boundary fixed points, Pacific J. Math. 222 (2005), 253–286.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M. D. Contreras, S. Diaz-Madrigal, and Ch. Pommerenke, Fixed points and boundary behaviour of the Koenigs function, Ann. Acad. Sci. Fenn. Math. 29 (2004), 471–488.MathSciNetzbMATHGoogle Scholar
  6. [6]
    P. L. Duren, Univalent Functions, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  7. [7]
    M. Elin and D. Shoikhet, Dynamic extension of the Julia-Wolff-Carathéodory theorem, Dynam. Systems Appl. 10 (2001), 421–437.MathSciNetzbMATHGoogle Scholar
  8. [8]
    M. Elin and D. Shoikhet, Angle distortion theorems for starlike and spirallike functions with respect to a boundary point, Int. J. Math. Math. Sci. 13 2006, Art. ID 81615.Google Scholar
  9. [9]
    M. Elin and D. Shoikhet, Linearization Models for Complex Dynamical Systems. Topics in Univalent Functions, Functional Equations and Semigroup Theory, Birkhäuser-Verlag, Basel, 2010.zbMATHGoogle Scholar
  10. [10]
    M. Elin, D. Shoikhet, and F. Yacobzon, A distortion theorem for functions convex in one direction, Complex Anal. Oper. Theory 5 (2011), 751–758.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. Elin, D. Shoikhet, and L. Zalcman, A flower structure of backward flow invariant domains for semigroups, Ann. Acad. Sci. Fenn. Math. 33 (2008), 3–34.MathSciNetzbMATHGoogle Scholar
  12. [12]
    J. B. Garnett and D. E. Marshall, Harmonic Measure, Cambridge Univ. Press, Cambridge, 2005.CrossRefzbMATHGoogle Scholar
  13. [13]
    W. K. Hayman, Subharmonic Functions, vol. 2, Academic Press, London, 1989.zbMATHGoogle Scholar
  14. [14]
    A. Lecko, On the class of functions convex in the negative direction of the imaginary axis, J. Aust. Math. Soc. 73 (2002), 1–10.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D. Minda, A reflection principle for the hyperbolic metric and applications to geometric function theory, Complex Variables Theory Appl. 8 (1987), 129–144.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    R. Nevanlinna, Analytic Functions, Springer-Verlag, New York-Berlin, 1970.CrossRefzbMATHGoogle Scholar
  17. [17]
    M. Ohtsuka, Dirichlet Problem, Extremal Length and Prime Ends, Van Nostrand, 1970.zbMATHGoogle Scholar
  18. [18]
    Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.CrossRefzbMATHGoogle Scholar
  19. [19]
    S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory, Academic Press, New York-London, 1978.zbMATHGoogle Scholar
  20. [20]
    T. Ransford, Potential Theory in the Complex Plane, Cambridge Univ. Press, Cambridge, 1995.CrossRefzbMATHGoogle Scholar
  21. [21]
    D. Shoikhet, Representations of holomorphic generators and distortion theorems for spirallike functions with respect to a boundary point, Int. J. Pure Appl. Math. 5 (2003), 335–361.MathSciNetzbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece

Personalised recommendations