Cauchy-Stieltjes Integrals And Areally Mean P-Valent Functions

In this thesis, we mainly consider Cauchy-Stieltjes integral of a complex Borel measure on the unit circle. Let F_α denote the function space consisting of these integrals, and let M_α denote the set of multipliers of F_α Our goal is to study the properties of F_α and some criterion conditions of M_α. The results obtained by us relate to areally mean p-valent function, logarithmic area, Taylor coefficient, self-similar set, and self-similar measure.In the first part, Cauchy-Stieltjes integrals and areally mean p-valent function will be consided. First, we give an equivalent definition of F_α, then estimate the logarithmic area for areally mean p-valent functions. Using this estimation we study the relation between this space and the logarithmic function of areally mean p-valent functions, finally discuss further this relation for the case of univalent functions. From this investigation, we obtain a growth estimation about the coefficients of bounded univalent functions, which is the best result up to now.In the second, combining Cauchy-Stieltjes integrals with fractal geometry, we discuss the Holder continuity of Cauchy-Stieltjes integrals of the self-similar measure, and give some criterion condition about wheather these integrals belong to M_α. To this end, we restrict our attention on the nonnegative Borel measures with uniformly β-dimension, and give the Holder continuity of Cauchy-Stieltjes integrals relevant to these measures. Finally, we character multipliers of F_α by using Taylor coefficients, and obtain some sufficient conditions for f ∈ M₀, which can't be improved.