Inner Functions

  • Javad Mashreghi
Part of the Fields Institute Monographs book series (FIM, volume 31)


The theory of Hardy spaces is a well established part of analytic function theory. Inner functions constitute a special family in this category. Therefore, it is natural to start with several topics on Hardy spaces and apply them in our discussions. However, we are not in a position to study this theory in detail and we assume that our readers have an elementary familiarity with this subject. In this chapter, we briefly mention, mostly without proof, the main theorems that we need in the study of inner functions. For a detailed study of this topic, we refer to [33].


Hardy Space Blaschke Product Singular Borel Measure Canonical Factorization Theorem Bergman Space 
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  1. 1.
    Ahern P (1979) On a theorem of Hayman concerning the derivative of a function of bounded characteristic. Pacific J Math 83(2):297–301MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ahern P, Clark D (1971) Radial nth derivatives of Blaschke products. Math Scand 28:189–201MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ahern P, Clark D (1974) On inner functions with H p-derivative. Michigan Math J 21:115–127MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ahern P, Clark D (1976) On inner functions with B p derivative. Michigan Math J 23(2):107–118MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Allen H, Belna C (1972) Singular inner functions with derivative in B p. Michigan Math J 19:185–188MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Belna C, Muckenhoupt B (1977) The derivative of the atomic function is not in B 2 ∕ 3. Proc Am Math Soc 63(1):129–130MathSciNetGoogle Scholar
  7. 7.
    Belna C, Colwell P, Piranian G (1985) The radial behavior of Blaschke products. Proc Amer Math Soc 93(2):267–271MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Beurling A (1948) On two problems concerning linear transformations in Hilbert space. Acta Math 81:239–255CrossRefGoogle Scholar
  9. 9.
    Blaschke W (1915) Eine erweiterung des satzes von vitali über folgen analytischer funktionen. Leipzig Ber 67:194–200Google Scholar
  10. 10.
    Bourgain J (1993) On the radial variation of bounded analytic functions on the disc. Duke Math J 69(3):671–682MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Carathéodory C (1929) Über die winkelderivierten von beschränkten analytischen funktionen. Sitzunber Preuss Akad Wiss 32:39–52Google Scholar
  12. 12.
    Cargo G (1961) The radial images of Blaschke products. J London Math Soc 36: 424–430MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Caughran J, Shields A (1969) Singular inner factors of analytic functions. Michigan Math J 16:409–410MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Cohn W (1983) On the H p classes of derivatives of functions orthogonal to invariant subspaces. Michigan Math J 30(2):221–229MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Collingwood EF, Lohwater AJ (1966) The theory of cluster sets. Cambridge tracts in mathematics and mathematical physics, No 56. Cambridge University Press, CambridgeGoogle Scholar
  16. 16.
    Colwell P (1985) Blaschke products. University of Michigan Press, Ann Arbor. Bounded analytic functionsGoogle Scholar
  17. 17.
    Cullen M (1971) Derivatives of singular inner functions. Michigan Math J 18:283–287MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Duren P, Schuster A (2004) Bergman spaces, volume 100 of mathematical surveys and monographs. American Mathematical Society, ProvidenceGoogle Scholar
  19. 19.
    Duren P, Romberg B, Shields A (1969) Linear functionals on H p spaces with 0 < p < 1. J Reine Angew Math 238:32–60MathSciNetzbMATHGoogle Scholar
  20. 20.
    Fatou P (1906) Séries trigonométriques et séries de Taylor. Acta Math 30(1):335–400MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Fricain E, Mashreghi J (2008) Integral means of the derivatives of Blaschke products. Glasg Math J 50(2):233–249MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Frostman O (1935) Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddel Lund Univ Mat Sem 3Google Scholar
  23. 23.
    Frostman O (1942) Sur les produits de Blaschke. Kungl. Fysiografiska Sällskapets i Lund Förhandlingar [Proc Roy Physiog Soc Lund] 12(15):169–182Google Scholar
  24. 24.
    Girela D, Peláez J, Vukotić D (2007) Integrability of the derivative of a Blaschke product. Proc Edinb Math Soc (2), 50(3):673–687Google Scholar
  25. 25.
    Hardy G, Littlewood J (1932) Some properties of fractional integrals. II. Math Z 34(1):403–439MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hedenmalm H, Korenblum B, Zhu K (2000) Theory of Bergman spaces, volume 199 of graduate texts in mathematics. Springer, New YorkCrossRefGoogle Scholar
  27. 27.
    Heins M (1951) A residue theorem for finite Blaschke products. Proc Amer Math Soc 2:622–624MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Herglotz G (1911) Über potenzreihen mit positivem, reellen teil in einheitskreis. S-B Sächs Akad Wiss Leipzig Math-Natur Kl 63:501–511Google Scholar
  29. 29.
    Julia G (1920) Extension nouvelle d’un lemme de Schwarz. Acta Math 42(1):349–355MathSciNetCrossRefGoogle Scholar
  30. 30.
    Linden C (1976) H p-derivatives of Blaschke products. Michigan Math J 23(1):43–51MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Lucas F (1874) Propriétés géométriques des fractionnes rationnelles. CR Acad Sci Paris 77:631–633Google Scholar
  32. 32.
    Mashreghi J (2002) Expanding a finite Blaschke product. Complex Var Theory Appl 47(3):255–258MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Mashreghi J (2009) Representation theorems in Hardy spaces, volume 74 of london mathematical society student texts. Cambridge University Press, CambridgeGoogle Scholar
  34. 34.
    Mashreghi J, Shabankhah M (2009) Integral means of the logarithmic derivative of Blaschke products. Comput Methods Funct Theory 9(2):421–433MathSciNetzbMATHGoogle Scholar
  35. 35.
    Nevanlinna R (1929) Über beschränkte analytische funcktionen. Ann Acad Sci Fennicae A 32(7):1–75Google Scholar
  36. 36.
    Plessner A (1923) Zur theorie der konjugierten trigonometrischen reihen. Mitt Math Sem Giessen 10:1–36Google Scholar
  37. 37.
    Privalov I (1918) Intégral de cauchy. Bulletin de l’Université, à SaratovGoogle Scholar
  38. 38.
    Privalov I (1924) Sur certaines propriétés métriques des fonctions analytiques. J de l’École Polytech 24:77–112Google Scholar
  39. 39.
    Protas D (1973) Blaschke products with derivative in H p and B p. Michigan Math J 20:393–396MathSciNetzbMATHGoogle Scholar
  40. 40.
    Riesz F (1923) Über die Randwerte einer analytischen Funktion. Math Z 18(1):87–95MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Riesz M (1931) Sur certaines inégalités dans la théorie des fonctions avec quelques remarques sur les géometries non-euclidiennes. Kungl Fysiogr Sällsk i Lund 1(4): 18–38MathSciNetGoogle Scholar
  42. 42.
    Rudin W (1955) The radial variation of analytic functions. Duke Math J 22:235–242MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Seidel W (1934) On the distribution of values of bounded analytic functions. Trans Am Math Soc 36(1):201–226MathSciNetCrossRefGoogle Scholar
  44. 44.
    Tsuji M (1959) Potential theory in modern function theory. Maruzen, TokyozbMATHGoogle Scholar
  45. 45.
    Vukotić D (2003) The isoperimetric inequality and a theorem of Hardy and Littlewood. Am Math Mon 110(6):532–536zbMATHCrossRefGoogle Scholar
  46. 46.
    Walsh JL (1939) Note on the location of zeros of the derivative of a rational function whose zeros and poles are symmetric in a circle. Bull Am Math Soc 45(6):462–470CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Javad Mashreghi
    • 1
  1. 1.Département de mathématiques et de statistiqueUniversité LavalQuébecCanada

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