الفهرس 
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Abstract
Geometric Function Theory is that branch of complex analysis, which deals and studies the geometric properties of the analytic functions. It is an area of Mathematics characterized by an intriguing marriage between geometry and analysis. Its origin date from the 19th century but new applications arise continually. Geometric Function Theory is a classical subject. Yet it continues to find new applications in an ever-growing variety of areas such as modern mathematical physics, more traditional fields of physics such as fluid dynamics, non-linear integrable systems theory of partial differential equations. The theory of univalent function is one of the most beautiful subjects in Geometric function Theory. Its origin ( a part from The Riemann mapping Theorem) can be traced to the 1907 paper of Koebe [72], to Gronwall’s proof of the Area Theorem in 1914 and Bieberbach’s estimate for the second coefficient of normalized univalent functions in 1916 and its consequences. By then, univalent function theory became a subject in its own right. Let ℂ be the complex plane and △={z:z∈C and |z|<1}be the open unit disc in ℂ.We denote by A the class of all normalized analytic functions in △.In this thesis, we obtain subordination properties, inclusion relations, coefficient bounds, convolution conditions, sufficient condition for starlikeness and convexity and Fekete-Szego type coefficient inequalities for various classes of analytic functions in the open unit disc △ and in the punctured unit disc △^*. Also we obtain bounds for the initial coefficients of the Taylor series expansion of several classes of bi-univalent functions.
The thesis consists of six chapters.
Chapter 1:This chapter is an introductory chapter contains basic concepts, definitions and preliminary results which are absolutely essential for understanding the results in subsequent chapters.Chapter 2 :This chapter consists of two sections :
Section 2.1. In this section, we introduce a new subclass, of starlike functions with respect to k-symmetric points of complex order γ(γ≠0)consists of functions f(z)∈A, which are analytic in △.Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some known results, and some other new results are obtained. Section 2.2.In this section, we introduce the new subclass (0≤α<1,0≤λ≤1,β≥0)β-U^~ CV^k (λ,α) of functions f(z)∈Awith negative coefficients,and determine coefficient estimates, neighborhoods and partial sums for functions f(z) belonging to this class. Chapter 3 : This chapter consists of two sections : Section 3.1. In this section, we introduce a new subclass of S^* (λ,,γ,M)(γ∈C^*=C⁄{0} ,m=1-1/M,M≥1,λ∈C)which unifies the classes of bounded meromorphicstarlike and meromorphic convex functions of complex order. Making use of Al-Oboudi operator a more general class S^* (n,λ,γ,M). Section 3.2. In this section, we study and investigate convolution properties, coefficient estimates and containment properties for the subclasses S^* (n,λ,γ,M).