Convolutions of harmonic convex mappings

Abstract

Abstract. The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right halfplane mapping or a normalized vertical strip mapping is convex in the direction of the real axis. provided that it is locally univalent. In this paper, we prove that in general the assumption of local univalency cannot be omitted. However, we are able to show that in some cases these harmonic convolutions are locally univalent. Using this we obtain interesting examples of univalent harmonic maps one of which is a map onto the plane with two parallel slits.

Construction of Subclasses of Univalent Harmonic Mappings

“…Clearly V ⊂ U. In [13], Goodman proved that U ⊂ S * and V ⊂ K. It is easy to see that U ⊂ R and V ⊂ W. In fact, if f ∈ U is given by (9), then…”

“…In the harmonic case, with f = h +ḡ and F = H +Ḡ belonging to H, their harmonic convolution is defined as f * F = h * H +g * G. Harmonic convolutions are investigated in [7,8,9,12,33]. Suppose that I and J are subclasses of H. We say that a class I is closed under convolution if I * I ⊂ I, that is, if f , g ∈ I then f * g ∈ I.…”

“…Since U and V are convex sets, therefore it suffices to show that the classes U and V are closed under convolution in view of Corollary 2.11. Let f , F ∈ V be given by (9). Then…”

“…The main crux of the proof relies on the observation that if f ∈ V is given by (9), then ∞ n=2 n 2 |a n | 2 ≤ 1. Since U and V are convex sets, therefore it suffices to show that the classes U and V are closed under convolution in view of Corollary 2.11.…”

A Note on Convexity of Convolutions of Harmonic Mappings

“…Theorem B ( [7]). Let f = h + g ∈ R 0 H with the dilatation ω(z) = e iθ z n , where n ∈ Z + and θ ∈ R. If n = 1, 2, then f 0 * f ∈ S 0 H and is convex in the direction of the real axis.…”