摘要
This problem was put forward by H.S.Shapiro on Bull,London Math.Soc.16(1984)490-517,numbered 6.88[1].The original description likes following;Let the function f(x)=z+a2x^w…in Smap the unit disc onto a domain with finite area A.Then Bieberbach's inequality |a2|≤2 can be sharpened to the following;|a2|≤2-CA^-1/2(*)Where C is absolute constant.What is the best value of C> Aharonov and Shapiro have shown that(*) holds for some C,and havea conjecture concerning the sharp constant C and the extremal function for(*).They also conjecture that |a2|≤2-C1l^-1,where l is the length of δf(D).(H.S.Shapiro) On condition that f belongs to star-alike family in addition to the above presumptions.I have resolved this conjecture and got a precise estimates;|a2|≤2(1-π^1/2A^-1/2)^1/2.The equation is met when f is and only is identical.
