The Goluzin Problem Of Univalent Functions

Content:The central problem of univalent functions theory is coefficient problem , and the difference of the moduli of adjacent coefficients is a unsolved problem which is difficult and interesting .Professor HU KE has studied this problem ,and obtain the answer on close-to-convex function.This paper studies the Goluzin problem of univalent functions .First, I generalize the class of close-to-convex functions to a larger class of functions--sc(α) .Let sc(α) be the class of functions f(z)which must satisfy the the conditions :f(z)∈S' , α∈R , (?)g(z)∈S* , make >o , where<1. When α =1 ,we have f∈sc. Because sc(1) = sc, we haveSc(α) Sc.Furthermore , this paper makes use of a lot of basis knowledges of univalent functions ,such as the Hergloz theorem of the positive real part functions , the theorem of the area , the inequality of Goluzin ,the inequality of Milin-Lebjev , the theorem of Milin ,the inequality of Schwarz ,the integration by parts ,and so on .With some new method of estimation ,1 obtain an estimation of thedifference or moduli or adjacent coemcients or ^ ; functions THEOREM :When we have . . n = 2,3......, where s aconstant which is only constrained by a , and the order --1 is thebest estimation.This theorem generalizes Professor HU KE 's theorem [1] on adjacent coefficients of close-to-convex functions .