On the coefficients of quasiconformality for convex functions

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.