Harmonic mappings and solutions of a differential equation related to the de la Vallee Poussin means

Analytic de la Vallee Poussin means have been studied for their geometric preserving properties. They most notably led to the Polya-Schoenberg conjecture. This thesis is primarily concerned with mapping problems of complex-valued functions. The topics will include geometric and subordination properties of these functions some of which are related to these means.; Although it is known that the Polya-Schoenberg conjecture does not extend to harmonic functions, we can still utilize these means to explore mapping problems for harmonic functions. We begin by developing a family of convex harmonic functions by way of the de la Vallee Poussin means and the Hadamard product. In addition, we discuss subordination as applied to harmonic functions and construct a subordination chain.; Next, geometric and subordination characteristics of solutions of a differential equation related to the de la Vallee Poussin means are explored. In doing so, we develop several families of analytic functions with properties similar to these means and produce several subordination chains.; In addition, the development of regular harmonic n -gon mappings results in an example of a subordination chain. This development provides other convex harmonic functions that can be related to repeated convolutions. We then connect the subordination of analytic functions to harmonic functions to establish coefficient bounds for harmonic mappings. Lastly, we briefly explore harmonic functions on the punctured unit disk.