Complex Analysis and Operator Theory

, Volume 10, Issue 1, pp 61–68 | Cite as

Phragmén–Lindelöf Principles for Generalized Analytic Functions on Unbounded Domains



We prove versions of the Phragmén–Lindelöf strong maximum principle for generalized analytic functions defined on unbounded domains. A version of Hadamard’s three-lines theorem is also derived.


Phragmén–Lindelöf principle Generalized analytic function Pseudoanalytic function Three-lines theorem 

Mathematics Subject Classification

30G20 30C80 



The authors are grateful to Joseph Burrier for his assistance. They also thank the referee for some useful comments.


  1. 1.
    Baratchart, L., Leblond, J., Rigat, S., Russ, E.: Hardy spaces of the conjugate Beltrami equation. J. Funct. Anal. 259(2), 384–427 (2010)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bers, L.: Theory of pseudo-analytic functions. Institute for Mathematics and Mechanics. New York University, New York (1953)Google Scholar
  3. 3.
    Carl, S.: A maximum principle for a class of generalized analytic functions. Complex Var. Theory Appl. 10(2–3), 153–159 (1988)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, S.-S.: On a class of quasiconformal functions in Banach spaces. Proc. Am. Math. Soc. 37, 545–548 (1973)MATHCrossRefGoogle Scholar
  5. 5.
    Fischer, Y., Leblond, J.: Solutions to conjugate Beltrami equations and approximation in generalized Hardy spaces. Adv. Pure Appl. Math. 2(1), 47–63 (2011)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Fischer, Y., Leblond, J., Partington, J.R., Sincich, E.: Bounded extremal problems in Hardy spaces for the conjugate Beltrami equation in simply-connected domains. Appl. Comput. Harmon. Anal. 31(2), 264–285 (2011)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Iwaniec, T., Martin, G.: Geometric function theory and non-linear analysis. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2001)Google Scholar
  8. 8.
    Krantz, S.G.: Geometric function theory. Explorations in complex analysis. Cornerstones. Birkhäuser Boston Inc., Boston (2006)Google Scholar
  9. 9.
    Kravchenko, V.V.: Applied pseudoanalytic function theory. In: With a Foreword by Wolfgang Sproessig. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2009)Google Scholar
  10. 10.
    Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, London (1939)MATHGoogle Scholar
  11. 11.
    Vekua, I.N.: Generalized analytic functions. Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc, Reading, Mass (1962)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.CNRS, UMR 5208, Institut Camille Jordan, Ecole Centrale de Lyon, INSA de Lyon, Université Lyon 1Université de LyonVilleurbanne CedexFrance
  2. 2.School of MathematicsUniversity of LeedsLeedsUK

Personalised recommendations