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Convolutions for special classes of harmonic univalent functions
Institution:Department of Mathematical Sciences, Kent State University 14111 Clairdon Troy Raod, Burton, OH 544201-9500, U.S.A.;College of Charleston 66 George Street, Charleston, SC 29424, U.S.A.
Abstract:Ruscheweyh and Sheil-Small proved the PólyarSchoenberg conjecture that the class of convex analytic functions is closed under convolution or Hadamard product. They also showed that close-to-convexity is preserved under convolution with convex analytic functions. In this note, we investigate harmonic analogs. Beginning with convex analytic functions, we form certain harmonic functions which preserve close-to-convexity under convolution. An auxiliary function enables us to obtain necessary and sufficient convolution conditions for convex and starlike harmonic functions, which lead to sufficient coefficient bounds for inclusion in these classes.
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