Properties Of Some Meromorphic Functions Related To Q-difference Operator

In recent years,q-calculus has been widely used in mathematics and physics as an important tool for studying quantum groups,q-deformed superalgebras,fractal and multifractal measures,optimal control problems and chaotic dynamical systems and so on.The q-difference operator,as the extension of derivative by q-calculus,is studied by many scholars by defining new function classes in combination with the subordination.It is of great research value and theoretical significance in geometric function theory,fractional differential equations,meromorphic functions,convolution theory and other fields.In 2014,H.Aldweby and M.Darus studied a class of meromorphically univalent functions referring to q-hypergeometric functions,and obtained some results on the partial sums of functions in this class.In 2016,T.M.Seoudy and M.K.Aouf defined the classes of q-starlike functions and q-convex functions of complex order with relation to q-difference operator(or q-derivative operator)in the open unit disk,and then they discussed the estimates of the second and third coefficients of these function classes under a certain relationship.In 2019,H.M.Srivastava and M.Tahir et al.defined the class of q-starlike functions with negative coefficients by combining q-difference operator with Janowski functions,and studied the distortion theorems and radii problems of this class.In this paper,a new subclass ?p,q(?,A,B)of multivalent meromorphic functions is defined by using Janowski functions and q-difference operator,and the sufficient and necessary conditions,coefficient estimates,growth and distortion theorems,radii of starlikeness and convexity,partial sums and closure theorems of some functions in this class are studied.Every meromorphic function can be expressed as the ratio of two analytic functions in their domain,where the denominator is not always zero.Accordingly,a new subclass Sp,q(?,A,B)of multivalent analytic functions is defined.The Fekete-Szego inequality and other properties of some functions in this class are discussed.This paper consists of the following three chapters.Chapter 1 is the introduction and preliminary knowledge,which mainly introduces the research background and current situation of this paper,as well as the basic knowledge required for the research,and defines the class Sp,q(?,A,B)and class ?p,q(?,A,B)based on the subordination.In Chapter 2,we deduce the Fekete-Szeg(?) inequality for functions in class Sp,q(?,A,B)by applying Schwarz function w(z).In the third chapter,we study the properties of functions in class Sp,q(?,A,B)and class?p,q(?,A,B),for instance,the sufficient and necessary conditions,coefficient estimates,growth theorems,distortion theorems,radii of starlikeness,radii of convexity,partial sums and closure theorems.