THE MINIMUM OF THE MEAN DIAMETERS OF THE SCHLICHT FUNCTIONS IN THE CLASS Σ

Let the function(?)be regular andschlicht in the domain l|ξ|∞.All such functions form a class Σ.Let Γ_F be the boundary of the image of|ξ|1 represented by w=F(ξ).Let α_1,α_2,…,α_n be n(nl)distinct points on Γ_F,P_n(F)the maximumof the quantity(?)as the points vary.The maximum problem of the mean diameter P_n(F)hasbeen considered by Bieberbach,Schiffer and Golusin[1][2].The aim ofthe present note is to discuss the minimum Q_n of P_n(F).Fixing a point α_1=a of Γ_F,the maximum of(l)will be denoted by P_n(F,a),when a_2,……,a_n,vary on Γ_F.The following result is attained.Theorem.There exists a function F_1 of Σ such that Q_n=P_n(F_1).Forthis extremal function F_1,P_n(F_1,a)is a constant for any point a on Γ_F_1.By means of this proposition,we can establish the followingCorollary.If Q_2=P_2(F_1).Then the boundary of the image W_F_1 of|ξ|1 mapped by F_1 is a closed convex curve.Moreover,there exist pointsexterior to the image W_F_1.