Some Applications Of Higher Order Derivatives Of Analytic Functions In Three Special Function Spaces

The study of analytic function spaces is one of the important fields of complex analysis,in which considering the distance from a point in some special function spaces to its subspaces is a hot issue in recent years.Jones,Ghatage,Zheng,Tjani,Zhao and others have studied the distance from a point in Bloch space to its subspaces and related problems,and achieved fruitful results.On the basis of previous studies,this paper will give a new characterization of the distance from Bloch space to F(p,p-2,s)space by the higher derivatives of analytic functions.The study of complex differential equations is also one of the hot issues in the field of complex analysis.In 1982,the famous mathematician Pommerenke creatively combined the function spaces with complex differential equations and considered the coefficient conditions that the solutions of second order complex linear differential equations belong to Hardy spaces.In recent years,three Finnish mathematicians Heittokangas,Korhonen and R?tty? have further generalized the research results of Pommerenke,considering the coefficient conditions that the solutions of higher order linear differential equations belong to some special spaces,such as weighted Hardy spaces,weighted Dirichlet spaces and so on.The study of the combination of complex differential equations and function spaces has gradually become one of the hot topics in modern complex analysis.In this paper,combined with previous methods,we will study the coefficient conditions when the solutions of higher order linear complex differential equations are found such that solutions belong to the HK2 space,Dpγ space specially.The results are mainly divided into two parts:using the higher order derivatives of analytic functions,we give a new characterization of the distance from Bloch space to F(p,p-2,s)space,and give the characterizations of higher order derivatives of points in HK2 space and Dpγ space.Using this result,we give the sufficient conditions such that all solutions of higher order linear differential equations belong to HK2 space,Dpγ space.The above content will be divided into six chapters.In Chapter 1,we introduce some background and main results of this paper.In Chapter 2,we introduce some basic knowledge:s-Carleson measure,HK2 space,Dpγspace,F(p,q,s)space,the distance from a point in Bloch space to its subspace,the boundedness,compactness on the composite operators,and the product of n-order differential operatorsIn Chapter 3,we give the proof of the higher order derivative characterization of the distance from Bloch space to F(p,p-2,s)space.In Chapter 4,we prove the boundedness,compactness of the product of composition operators and n-order differential operators from Bloch spaces to the closure of F(p,p-2,s)space.In Chapter 5,we give the proofs of the higher derivative characterization of HK2 space,Dpγ space.With the characterization of higher order derivatives in HK2 space and Dpγ space,we can study the properties of the solutions of higher order linear differential equations.In Chapter 6,we give the proof of the sufficient conditions that the solutions of the linear differential equations belongs to the HK2 space,Dpγ space.