| In the 1920s, the study of complex valued harmonic functions of one complex variable was motivated by their connection with minimal surfaces. It is only recently that harmonic functions have been studied as generalizations of conformal mappings. Clunie and Sheil-Small initiated that approach in 1984, and their influential paper has sparked new growth in the field. The driving force of much of the recent research is to examine what properties of analytic functions generalize to harmonic functions. Although there are many natural similarities with analytic functions, there are counterintuitive results, particularly in connection with the boundary behavior of harmonic maps.;Viewing harmonic functions as a generalization of analytic functions stems from the crucial fact that any harmonic function in the open unit disk can be written as a sum of an analytic and the conjugate of an analytic (anti-analytic) function. This thesis concentrates on finding the connections between the analytic and geometric properties of harmonic mappings. The main results lie in four broad categories: invariant families of functions, subordination for harmonic functions, geometric properties of a special class of functions called "shears," and constructing new examples of harmonic functions.;We start in Chapter 2 by extending the ideas of Pommerenke and Sheil-Small to explore affine and linear invariant families of harmonic functions. In Chapter 3, we initiate the systematic study of subordination of harmonic functions and extend various basic results from the analytic to the harmonic setting, including results that are related to invariant families. Chapters 4 and 5 explore the "shear" construction, which constructs functions convex in one direction. Chapter 4 deals with boundary behavior of shear functions, and Chapter 5 solves several extremal problems for shears. Chapter 6 introduces a new method for beginning with a univalent analytic function and constructing a univalent harmonic function. |
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