# On the coefficients of quasiconformality for convex functions

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## Abstract

Let . In this paper the authors computed the quantity

*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$$
k_f \left( r \right) = \mathop {\sup }\limits_{z \in f\left( {rE} \right)} \left| {\frac{{\tfrac{{\partial ^2 R\left( {f\left( {rE} \right),z} \right)}}
{{\partial \bar z^2 }}}}
{{\tfrac{{\partial ^2 R\left( {f\left( {rE} \right),z} \right)}}
{{\partial z\partial \bar z}}}}} \right|
$$

*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

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## References

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*Lectures on quasiconformal mappings*, (D. Van Nostrand Comp., Princeton, New Jersey, 1966; Mir, Moscow, 1969).MATHGoogle Scholar - 2.F. G. Avkhadiev and K.-J. Wirths,
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© Pleiades Publishing, Ltd. 2010