Hadamard Products of Convex Harmonic Mappings

ISSN： 0278-1077

Source： Complex Variables, Vol.47, Iss.2, 2002-02, pp. : 81-92

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Abstract

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + overline {g} $ where h and g are analytic functions. If $ f in K_H $ has $ h(0) = 0, g(0) = 0, h'(0) = 1$, and $ g'(0) = 0 $, then $ f in K_H0 $. For $ f in K_H0 $ and phiv analytic in the unit disk, an integral representation for $ ftilde {*}varphi = h*varphi + overline {g*varphi } $ is found. With phiv a strip mapping, $ ftilde {*}varphi $ is shown to be in $ K_H0 $. In a 1958 paper, Pólya 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. It is known that the analogue of that result for harmonic mappings is false. In this paper, 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 $ f_n in K_H0 $ which take the unit disk to regular n-gons.