Mapping Properties of Some Harmonic Functions
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초록
Let D denote the open unit disc centered at the origin in the complex plane, and let S denote the usual class of normalized univalent functions defined on D. Here ’*’ denotes the Hadamard product (or convolution) of two analytic functions. We find conditions on f such that Mc,h[f] is harmonical convex; and conditions on h and f such that Mc,h[f] is stably harmonically convex. We also investigate properties of the classical Cesaro means and de la Vallee Poussin means of Mc,h[I], where I(z) = z/(1-z). Furthermore, we present, among other results, a sub ordination relation between the Cesaro means and the de la Vallee Poussin means of a harmonic function.
Abstract
Let D denote the open unit disc centered at the origin in the complex plane, and let S denote the usual class of normalized univalent functions defined on D. Here ’*’ denotes the Hadamard product (or convolution) of two analytic functions. We find conditions on f such that Mc,h[f] is harmonical convex; and conditions on h and f such that Mc,h[f] is stably harmonically convex. We also investigate properties of the classical Cesaro means and de la Vallee Poussin means of Mc,h[I], where I(z) = z/(1-z). Furthermore, we present, among other results, a sub ordination relation between the Cesaro means and the de la Vallee Poussin means of a harmonic function.