Applications Of Differential Subordinations And Differential Superordinations In The Integral Operators

The studies of the properties and applications of integral operators with differential subordination and differential superbordination play an important role in the analytic function theory.Differential subordination and differential superordination have been widely applied in the research of differential equations,meromorphic functions,Banach spaces and multiple complex functions.Therefore,it has important theoretical significance and practical application value for the study of the dependent relation and the integral operator problem.In this paper,we mainly study the properties of integral operators by using differential subordination and differential superordination in analytic functions.This paper mainly studies the following three aspects:Firstly,on the properties of integral operators of several classes of meromorphic multivalent functions;such as inclusion relations,integral-preserving,subordinate relationships.Secondly,third order differential subordinates and super subordinates related to generalized Hurwitz-Lerch Zeta functions;Thirdly,the related properties of a new subclass of analytic functions generated by univalent harmonic mappings,such as subordinate relationships,integral representations,coefficient inequality,and convolution properties.For a detailed of differential subordination and differential superordination,the main works of the study are as follows:1.By using differential subordination,the paper will define a new operator J_?,p,?^n,l and study the inclusion relations,integral-preserving properties of the subclasses of meromorphic multivalent functions related to these operators,and studies the subordination and superordination results related to these operators.Furthermore,some of the double subordination properties are derived.2.From the basic theory of function p that satisfy third-order differential superordination condition in the unit disk,by choosing suitable classes of admissible functions,the paper will derive some third-order differential subordination and differential subordination results for meromorphic functions,which are defined by using the operator W_s,bf?z?involving Hurwitz-Lerch Zeta function.3.Concerning the harmonic mapping coefficients related to classical Bieberbach conjecture,the paper will define a new class F???.The paper will discuss subordination property,integral representation,coefficient inequalities and convolution property for the function class F???.Some of the results present here would be seen as further discussion of earlier works,at the same time,the results we obtain are new and important.