Lobachevskii Journal of Mathematics

, Volume 31, Issue 4, pp 323–325

# 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

## References

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L. Ahlfors, Lectures on quasiconformal mappings, (D. Van Nostrand Comp., Princeton, New Jersey, 1966; Mir, Moscow, 1969).
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L. A. Aksent’ev and A. N. Akhmetova, On Mappings related to the gradient of the conformal radius, Math. Notes 87.1, 3 (2010).
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C. Carathéodory, Theory of functions of a complex variable II (Chelsea Pub. Comp., New York, 1960).Google Scholar