丘成桐大学生数学竞赛试卷

S.-legeStudentMathematicsContests2010AnalysisandDifferentialEquationsTeam(Pleaseselect5problemstosolve)1.a)Letf(z)beholomorphicinD:|z|<1and|f(z)|≤1(z∈D).Provethat|f(0)|−|z||f(0)|+|z|≤|f(z)|≤.(z∈D)1+|f(0)||z|1−|f(0)||z|b)Foranyfinitecomplexvaluea,provethat