Advertisement

Lobachevskii Journal of Mathematics

, Volume 31, Issue 4, pp 323–325 | Cite as

On the coefficients of quasiconformality for convex functions

Article

Abstract

Let f be holomorpic and univalent in the unit disc E and f(E) be convex. We consider the conformal radius R = R(D, z) = |f′(ζ)|(1−ζ \( \bar \zeta \)) of D = f(E) at the point z = f(ζ). In [3] and [4] the coefficient k f (r), r ∈ (0, 1), of quasiconformality has been defined by the equation
$$ k_f \left( r \right) = \mathop {\sup }\limits_{z \in f\left( {rE} \right)} \left| {\frac{{\tfrac{{\partial ^2 R\left( {f\left( {rE} \right),z} \right)}} {{\partial \bar z^2 }}}} {{\tfrac{{\partial ^2 R\left( {f\left( {rE} \right),z} \right)}} {{\partial z\partial \bar z}}}}} \right| $$
. In this paper the authors computed the quantity k f (r) for some convex functions. These examples led them to the conjecture that k f (r) ≤ r 2 for any convex function holomorphic in E. The function f(ζ) = log((1 + ζ)/(1 −ζ)), which was among their examples, shows that this bound is sharp for any r∈ (0, 1). In the present article, we will prove that the above conjecture is true and that the the above example is essentially the only one for which equality is attained.

Keywords and phrases

Convex functions conformal radius coefficient of quasiconformality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Ahlfors, Lectures on quasiconformal mappings, (D. Van Nostrand Comp., Princeton, New Jersey, 1966; Mir, Moscow, 1969).MATHGoogle Scholar
  2. 2.
    F. G. Avkhadiev and K.-J. Wirths, The conformal radius as a function and its gradient image, Israel J. Math. 145, 349 (2005).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    L. A. Aksent’ev and A. N. Akhmetova, Mappings connected with the gradient of conformal radius, Russ. Math. 53.6, 49 (2009).CrossRefMathSciNetGoogle Scholar
  4. 4.
    L. A. Aksent’ev and A. N. Akhmetova, On Mappings related to the gradient of the conformal radius, Math. Notes 87.1, 3 (2010).CrossRefGoogle Scholar
  5. 5.
    C. Carathéodory, Theory of functions of a complex variable II (Chelsea Pub. Comp., New York, 1960).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Depart. of Mech. and Math.Kazan UniversityKazanRussia
  2. 2.Institut für Analysis and Algebra, TU BraunschweigBraunschweigGermany

Personalised recommendations