Properties Of Some Analytic Functions Associated With Q-Derivative Operator

In the development of geometric function theory,q-derivative is an important research tool.It has been widely used in many fields such as optimal control,ordinary fractional calculus,q-integral equations and q-transformation analysis.Recently,some scholars have defined new classes of analytic functions by combining q-derivative with differential subordination,and studied their geometric properties.This is of great significance for promoting the development of geometric function theory.In 2019,H.M.Srivastava and Q.Z.Ahmad et al.used convolution to generalize q-Ruscheweyh integral operator,and studied the coefficient estimates,Hankel determinant and Toeplitz determinant for a subclass of q-starlike functions associated with a general conic domain.In 2021,M.S.Ur.Rehman and Q.Z.Ahmad et al.defined a new class of q-starlike functions by using q-derivative and Janowski functions,and studied the Fekete-Szego inequality and several sufficient conditions.In 2022,G.Murugusundaramoorthy and Luminita-Ioana Cotirla studied the coefficient estimates and Fekete-Szego inequality for bi-univalent functions associated with Legendre polynomial.In this paper,we define new classes IΠq[A,B,λ,β]of analytic functions and STq[k,λ,β]of starlike functions by using q-derivative operator and differential subordination,and study some properties of functions in these classes.We also define a new subclass IΣq[φ,λ,β]of bi-univalent functions using Legendre polynomial,and study the coefficient estimates and Fekete-Szego inequality.This paper is made up of the following four chapters.In Chapter 1,we introduce the background and current situation of this paper,as well as some theoretical knowledge needed for the study,and define the classes IΠq[A,B,λ,β]、STq[k,λ,β]and IΣq[φ,λ,β]based on q-derivative operator and differential subordination.The second chapter is the Fekete-Szego inequality and several sufficient conditions for functions in the class IΠq[A,B,λ,β]by using the Schwarz function w(z).In Chapter 3,we discuss the coefficient estimates,Hankel determinant and Toeplitz determinant of functions in class STq[k,λ,β].In the fourth chapter,we study the coefficient estimates and Fekete-Szego inequality of functions in class IΣq[φ,λ,β]by applying Legendre polynomial.