Estimates of the Derivatives of Meromorphic Maps from Convex Domains into Concave Domains

  • Farit G. AvkhadievEmail author
  • Karl-Joachim Wirths


Let ω be a proper convex subdomain of the complex plane C. Let further π1 ⊂ ℂ be a compact convex set containing more than one point and π = \({\rm}\overline C\ \Pi_1\). We denote by R Ω (z) and R π (ω) the conformal radius of Ω at z and of π at finite points ω, respectively. We are concerned with the set A(Ω, π) of functions f: Ω → π meromorphic on Ω. We prove that for n ≥ 2, fA(Ω, π), z∈Ω and f(z) finite the inequalities
$${|f^{(n)}(z)|\over n!}{(R_\Omega(z))^n\over R_\Pi(f(z))} \leq {(1+p)^{n-2}\over p^{n-1}} {\sum^n_{k=0}} p^k$$
are valid, where p is a measure for the distance between f(z) and the point at infinity.

We give examples showing that equality is possible in this estimate.


Convex domain concave domain nth derivative conformal radius subordination 

2000 MSC

30C80 30C55 30C20 


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  1. 1.
    F. G. Avkhadiev, Ch. Pommerenke and K.-J. Wirths, On the coefficients of concave univalent functions, Math. Nachr. 271 (2004), 3–9.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    F. G. Avkhadiev and K.-J. Wirths, Convex holes produce lower bounds for coefficients, Complex Variables 47 (2002), 553–563.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    F. G. Avkhadiev and K.-J. Wirths, Schwarz-Pick inequalities for derivatives of arbitrary order, Constr. Approx. 19 (2003), 265–277.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    F. G. Avkhadiev and K.-J. Wirths, Punishing factors for angles, Comp. Methods Funct. Theory 3 (2003), 127–141.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    F. G. Avkhadiev and K.-J. Wirths, Schwarz-Pick inequalities for hyperbolic domains in the extended plane, Geom. Dedicata 106 (2004), 1–10.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    F. G. Avkhadiev and K.-J. Wirths, Punishing factors for finitely connected domains, Monatshefte f. Math. 147 (2006), 103–115.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    F. G. Avkhadiev and K.-J. Wirths, Sharp bounds for sums of coefficients of inverses of convex functions, Comp. Methods Funct. Theory, 7 (2007), 105–109.MathSciNetzbMATHGoogle Scholar
  8. 8.
    —, The punishing factors for convex pairs are 2nt−1, to appear in Revista Mat. Ib eroamericana.Google Scholar
  9. 9.
    F. G. Avkhadiev and K.-J. Wirths, A proof of the Livingston conjecture, Forum Math. 19 (2007), 149–157.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    F. G. Avkhadiev and K.-J. Wirths, Subordination under concave univalent functions, Complex Var. Elliptic Equ. 52 (2007), 299–405.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    —, Punishing factors and Chua’s conjecture, to appear in Bull. Belg. Math. Soc., Simon Stevin.Google Scholar
  12. 12.
    L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137–152.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    K. S. Chua, Derivatives of univalent functions and the hyperbolic metric, Rocky Mountain J. Math. 26 (1996), 63–75.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. L. Duren, Theory of Hp Spaces, Academic Press, New York, 1970.zbMATHGoogle Scholar
  15. 15.
    L. Fejér, Über gewisse durch die Fouriersche und Laplacesche Reihe definierten Mittelkurven und Mittelflächen, Palermo Rend. 38 (1914), 79–97.CrossRefzbMATHGoogle Scholar
  16. 16.
    P. Henrici, Applied Computational Complex Analysis, Vol. 1, Wiley, New York, 1974.zbMATHGoogle Scholar
  17. 17.
    Z. J. Jakubowski, On the upper bound of the functional \(|f^{(n)}(z)| (n = 2,3,...)\) in some classes of univalent functions, Ann. Soc. Math. Polon. Ser. I: Comment. Math., 17 (1973), 65–69.MathSciNetzbMATHGoogle Scholar
  18. 18.
    J. A. Jenkins, On a conjecture of Goodman concerning meromorphic univalent functions, Michigan Math. J. 9 (1962), 25–27.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    E. Landau, Einige Bemerkungen über schlichte Abbildung, Jber. Deutsche Math. Verein. 34 (1925/26), 239–243.Google Scholar
  20. 20.
    A. E. Livingston, Convex meromorphic mappings, Annales Pol. Math. 59 (1994), 275–291.MathSciNetzbMATHGoogle Scholar
  21. 21.
    A. Marx, Untersuchungen über schlichte Abbildungen, Math. Ann. 107 (1932/33), 40–65.MathSciNetCrossRefGoogle Scholar
  22. 22.
    J. Miller, Convex and starlike meromorphic functions, Proc. Amer. Math. Soc. 80 (1980), 607–613.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ch. Pommmerenke, Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 1975.Google Scholar
  24. 24.
    —, personal communication, December 3, 2002.Google Scholar
  25. 25.
    M. S. Robertson, Quasi-subordination and coefficient conjectures, Bull. Amer. Math. Soc. 76 (1970), 1–9.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    St. Ruscheweyh, Uber einige Klassen im Einheitskreis holomorpher Funktionen, Ber. d. math.-stat. Sektion im Forschungszentrum Graz 7 (1974), 1–12.Google Scholar
  27. 27.
    St. Ruscheweyh, Two remarks on bounded analytic functions, Serdica, Bulg. Math. Publ. 11 (1985), 200–202.MathSciNetzbMATHGoogle Scholar
  28. 28.
    T. Sheil-Small, On the convolution of analytic functions, J. Reine Angew. Math. 258 (1973), 137–152.MathSciNetzbMATHGoogle Scholar
  29. 29.
    E. Strohhäcker, Beiträge zur Theorie der schlichten Funktionen, Math. Z. 37 (1933), 356–380.MathSciNetCrossRefGoogle Scholar
  30. 30.
    O. Szász, Ungleichungen für die Koeffizienten einer Potenzreihe, Math. Z. 1 (1918), 163–183.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    O. Szász, Ungleichheitsbeziehungen für die Ableitungen einer Potenzreihe, die eine im Einheitskreis beschränkte Funktion darstellt, Math. Z. 8 (1920), 303–309.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    K.-J. Wirths, On the residuum of concave univalent functions, Serdica Math. J. 32 (2006), 209–214.MathSciNetzbMATHGoogle Scholar
  33. 33.
    S. Yamashita, La dérivée d’une fonction univalente dans une domaine hyperbolique, C. R. Acad. Sci. Paris Ser I. Math. 314 (1992), 45–48.MathSciNetzbMATHGoogle Scholar
  34. 34.
    S. Yamashita, Localization of the coefficient theorem, Kodai Math. J. 22 (1999), 384–401.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    S. Yamashita, Higher derivatives of holomorphic functions with positive real part, Hokkaido Math. J. 29 (2000), 23–36.MathSciNetzbMATHGoogle Scholar

Copyright information

© Heldermann  Verlag 2008

Authors and Affiliations

  1. 1.Chebotarev Research InstituteKazan State UniversityKazanRussia
  2. 2.Institut für Analysis und AlgebraTU BraunschweigBraunschweigGermany

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