| Let f= u+iv be a two-times continuously differentiable complex-valued function in a complex plane domain Ω(?) C, where u(x,y) and u(x, y) are two variables real functions. If u and v satisfy the Laplace differentiable equations: Δu= Δv= O, then/is said to harmonic mapping in domain Ω, where Δ denotes the complex value Laplace operator:The main aim of this thesis is to study some properties of harmonic mappings in the plane, such as coefficient estimates, univalent radius, convolution, convex combinations, inclusion relationships, starlikeness and convexity. This thesis con-sists of five chapters and the arrangement is as follows.In Chapter one, we provide the background on our research and the statement of our main results, furthermore, we introduce some notations and concepts.In Chapter two, we investigate the properties of a subclass of close-to-convex harmonic mappings defined in the unit disk. Firstly, we give coefficient estimates, and obtain the upper bound of the Fekete-Szego problem, for this class of mappings. Secondly, we determine the radii of close-to-convex for the partial sum of this class. The obtained results improve the corresponding known ones. In particular, for some function subclasses, we solve the radii problems related to starlikeness and convexity. Finally, we consider growth, covering and area theorems of the class in the unit disk.In Chapter three, we study the convolutions of the harmonic mappings f0 and f in the unit disk, where f0 is the normalized right half-plane harmonic mappings, and f belong to certain subclasses of the harmonic mappings. Firstly, we consider the convolutions f0*f, where the dilation of/is the conformal itself mapping in the unit disk, and obtain a sufficient condition for the convolutions of such mappings to be convex in the direction of the real axis, and we note that the obtained results improve the corresponding known ones. Secondly, we introduce the generalized right half-plane harmonic mappings and vertical strip harmonic mappings, and we obtain the some sufficient conditions for the convolutions of such mappings and f0 to be convex in the direction of the real axis. Finally, we discuss the coefficient conditions and the radius problems of the convolutions fo*f, which belong to the subclasses of harmonic starlike or convex mappings, where f denotes the generalized harmonic right half-plane mappings.In Chapter four, we investigate the preserving properties of the convex combi-nations for some subclasses of harmonic mappings. Firstly, we introduce a subclass of harmonic quasiconformal mappings defined in the unit disk, and the sufficien-t conditions for the convex combinations of mappings in such classes to be in a similar class, and convex in a given direction, are established. In particular, we prove that the images of convex combinations in this class, for special choices of parameters, are convex. Secondly, we prove that the convex combinations of s-lanted half-plane harmonic mappings are convex harmonic mappings. At last, we introduce a class of harmonic mappings obtained by horizontal shearing slit confor-mal mappings, the sufficient conditions for the convex combinations of harmonic mappings of this family to be univalent and convex in the direction of the real axis are derived.In Chapter five, we investigate the properties of meromorphic analytic func-tions defined in the punctured unit disk. Firstly, we introduce a subclass of mero-morphic starlike functions in the unit disk. Some interesting results concerning subordination properties, integral representations, convolution characterizations, inclusion relationship and coefficient inequalities for the functions of this class are derived. Furthermore, we solve radius problems for certain related classes of mero-morphic strongly starlike functions and meromorphic parabolic starlike functions. Secondly, making use of a differential operator, we define some new subclasses of meromorphic multivalent functions and investigate their inclusion relationships, integral preserving and convolution properties. |
