【作者】 刘静；

【导师】 王仙桃；

【作者基本信息】 湖南师范大学 ， 基础数学， 2022， 硕士

【摘要】 设D表示复平面C上的子域,f是D上的复值函数.若f是二次连续可微的,且满足Laplace方程Δf=0,则称f是D上的调和映射;若F是D上的四次连续可微复值函数,且满足双Laplace方程Δ(ΔF)=0,则称F是D上的双调和映射,其中Δ表示Laplace算子(?)以及z=x+iy.众所周知,解析函数是复分析中的主要研究对象,而调和映射是解析函数的推广,双调和映射又是调和映射的推广.特别是双调和映射,关于此类映射的研究起源于力学、生物学等学科中的一些实际问题.因此,关于这些映射的研究得到了人们的极大关注.本学位论文利用微分算子引入了新的调和映射类、双调和映射类及其它们的子类,并对其进行了研究,得到了一些相关性质.本学位论文由三章构成,具体安排如下.在第一章中,我们简述了本学位论文所研究问题的背景,并陈述了所得到的主要结果.在第二章中,我们利用Salagean算子引入了两个调和映射类HNm,n(α,β)、(?),以及它们的子类(?);得到了这些类基于系数的充分条件和它们依赖于参数的包含关系;建立了它们中元素的偏差定理、以及关于凸组合的封闭性;找到了(?)中元素的具体表示形式.在第三章中,我们利用L算子引入一类双调和映射GBH(λ,α,ρ)及其子类TGBH(λ,α,ρ);得到了这些类基于系数的充分条件、充分必要条件;建立了它们中元素的偏差定理、关于凸组合的封闭性;证明了极值点的存在性,并得到了它们的具体表示形式.最后,我们还讨论了子类TGBH(λ,α,ρ)关于卷积运算的封闭性.更多还原

【Abstract】 Let D denote a subdomain of the complex plane C,and f be a complexvalued function on D.If f is second order continuously differentiable and satisfies the Laplace equation Δf=0,then f is said to be a harmonic mapping on D.If F is a four times order continuously differentiable complex-valued function on D and satisfies the double Laplace equation Δ(ΔF)=0,then F is said to be a biharmonic map on D,where Δ represents the Laplace operator(?) and z=x+iy.As we know,analytic functions are the most important research objects in complex analysis,the harmonic mappings are generalizations of analytic functions,and biharmonic mappings are generalizations of harmonic mappings.In particular,the motivation on the research of biharmonic mappings comes from practical problems in mechanics,biology etc.Therefore,the research on these mappings has received much attention.In this dissertation,by using two differential operators,several new harmonic mapping classes,biharmonic mapping classes and their subclasses are introduced,and their related properties are investigated.This dissertation consists of three chapters,and its arrangement is as follows.In the first chapter,the background of the research problems in this dissertation is briefly described,and the main results are stated.In the second chapter,firstly,by using the Salagean operator,two new harmonic mapping classes HNm,n(α,β),(?),and their subclasses(?)are introduced and then,a sufficient condition is obtained,which is based on their coefficients;an inclusion relation depending on the related parameters is obtained;a distortion theorem is established;the invariant under the convex combination is discussed,and the general expression of the elements in(?)(α,β)is obtained.In the third chapter,firstly,by using the L operator,a class of biharmonic mappings GBH(λ,α,ρ)and its subclass TGBH(λ,α,ρ)are introduced,and then,a sufficient condition and a necessary and sufficient condition for a mapping to belong to these classes are obtained,which are based on their coefficients;a distortion theorem is proved;the invariant under the convex combination is discussed,and the existence and the constitution of their extreme points are established.Finally,the invariant of TGBH(λ,α,ρ)under the convolution operation is demonstrated.更多还原