ON THE COEFFICIENTS OF QUASICONFORMALITY FOR CONVEX FUNCTIONS F. G. AVKHADIEV AND K.-J. WIRTHS 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 ζ ) of D = f (E) at the point z = f (ζ ). In [3] the coefficient k f (r),r ∈ (0, 1), of quasiconformality has been defined by the equation k f (r)= sup z∈f (rE)      ∂ 2 R(f (rE),z) ∂ z 2 ∂ 2 R(f (rE),z) ∂z∂ z      . 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. 1. Introduction Let D be a simply connected proper subdomain of C. The conformal radius R(D,z) of D at the point z ∈ D may be defined by the formula R(D,z)= |f ′ (ζ )|(1 − ζ ζ ), where f is a conformal mapping of the unit disc E onto D and z = f (ζ ). In [2] the authors of the present article proved some properties of the function ∇R(D,z)=2 ∂R(D,z) ∂ z , z ∈ D, for several types of domains. Among others, in [3] the authors considered the com- plex dilation (1) µ F (z)= ∂F (z) ∂ z ∂F (z) ∂z for the function F (z)= ∇R(D,z), compare [1] for the rˆole that this quantity plays in the theory of quasiconformal mappings. In [3], for r ∈ (0, 1) the coefficient k f (r) of quasiconformality for the conformal map f has been defined by the equation Date : May 13, 2010; File: AKS.tex. 2000 Mathematics Subject Classification. 30C45, 30C62. Key words and phrases. Convex functions, conformal radius, coefficient of quasiconformality. During the work on this article F. G. Avkhadiev was supported by a grant of the Deutsche Forschungsgemeinschaft. 1