| Functions f in the class KH are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as f = h + g¯ where h and g are analytic functions. If f ∈ KH has h(0) = 0, g(0) = 0, h' (0) = 1, and g'(0) = 0, then f ∈ K0H . Properties of the product f*&d5;4=h*4+g*4 of a harmonic function f = h + g¯ and an analytic function 4 are examined. For f ∈ K0H and 4 analytic in the unit disk, an integral representation for the product f*4&d5; is found. With 4 a strip mapping, f*4&d5; is shown to be in K0H . In a 1958 paper, Polya and Schoenberg conjectured that if f and g are conformal mappings of the unit disk onto convex domains, then the Hadamard product f * g of f and g has the same property. Ruscheweyh and Sheil-Small proved the conjecture for analytic mappings, and it is known that the analogue of that result for harmonic mappings is false. In this dissertation, some examples are given in which the property of convexity is preserved for Hadamard products of certain convex harmonic mappings. In addition, an integral formula is used to determine the geometry of the Hadamard product from the geometry of the factors. This is true in particular for the convolution of strip mappings with certain functions fn ∈ K0H which take the unit disk to regular n-gons. |
