Some Problems Related To Bounded Analytic Functions And Cantor Boundary Behavior

The pointcut of interdisciplinary research of complex analysis and fractal geometry is the study of Cauchy transform of self-similar measure, which also forced the study of Cantor boundary behavior (CBB) of analytic functions. The concept of CBB was proposed by professor Dong and Lau, which is closely related with Topology and Measure theory. Its study add vitality for classical complex analysis and open up a new direction for fractal geometry research.In this thesis, we select topics from above fields and mainly consider two problems:1. The CBB of Cauchy transform of the3-level Sierpinski gasket.2. The boundary behavior of the proper holomorphic mappings on bounded simply connected domains.Naturally, we split the thesis into four Chapters:Chapter1is introduc-tion; Chapter2is preliminary; In Chapter3, we study the Cauchy transform F on the3-level Sierpinski gasket K. In this chapter, we obtain the estimation of F near some boundary points of the unbounded connected component Δo of C\K and study the asymptotic growth of F’near these points. Making use of these results, we prove that F has Cantor boundary behavior (CBB) on Δo.In addition, we consider the Laurent expansion of F and figure out the recursion formula and asymptotic formula of the Laurent coefficients; In chapter4, we study the proper holomorphic mappings on bounded simply con-nected domains. In this chapter, by lifting simply connected domain onto unit disc, we obtain the equality relationship between the component numbers of the boundary point of the range and its different inverse image points.we also discuss the inequality relationship between the topological degree of the proper holomorphic mapping and the numbers of the different inverse image points based on the boundary point is a cut point or not.