Inclusion Properties Of Classes Of Multivalent Analytic Functions

Ch.Pommerenke introduced some special univalent functions defined by simple geometry property in univalent functions such as starlike function,convex function and close-convex function,etc..Afterwards,several authors studied univalent or multivalent function class(for example,starlike and strongly starlike function of order a,convex and strongly convex function of order ?).In 1973,Janowsk used Schwarz function to define subordination relation,and used the subordination relation to define a series of subclasses of analytic functions.In addition,several scholars also used the subordination and operators to define different function classes and studied the related properties of these functions classes.In 2004,Nak Eun Cho and Oh Sang Kwon studied inclusion properties for certain classes of multivalent functions associated with a family of linear operators.In 2011,H.M.Srivastava and S.M.Khairnar studied analytic function class defined by an integral operator involving the hypergeometric functions and discussed their interesting properties.In 2016,S.Devi,H.M.Srivastava and A Swaminathan studied inclusion properties of a class of functions involving the Dziok-Srivastava operator.Motivated by their work,the present paper uses generalized Bessel function to define a linear operator,it introduces some multivalent analytic function classes in terms of differential subordination,and investigate their inclusion property.Furthermore,we present some special cases and consequences of those theorems.The paper consists of two parts:The first part is introduction and preliminary results.It introduces the basic concepts,lemmas and defines some multivalent analytic function classes.The second part discusses various inclusion relation and argument properties of multivalent analytic function classes.Also,we present some special cases of those results.