Convolutions of Harmonic Mappings Convex in One Direction

Abstract

Let $$f_1$$ be a fixed univalent and sense-preserving harmonic mapping which is convex in the real direction but is not starlike in the unit disk $$|z|<1$$ . The convolution of $$f_1$$ with other harmonic mappings, eg. half-plane mappings, is not necessarily univalent in $$|z|<1$$ . In this paper, under suitable restriction on the dilatation of $$f$$ , we show that the convolutions of $$f_1$$ with certain slanted half-plane harmonic mappings $$f\in {\mathcal S^0}(H_{\gamma })$$ are necessarily convex in a direction. In addition, we consider a fixed harmonic mapping $$f_0$$ and $$f=h+\overline{g}\in {\mathcal S^0}(H_{\gamma })\cup {\mathcal S^0}(\Omega _\alpha )$$ with the dilatation $$\omega (z)=az^n$$ , where $${\mathcal S^0}(\Omega _\alpha )$$ denotes the class of asymmetric vertical strip mappings. We find the relationship between $$a$$ and $$n$$ such that $$f_0*f$$ is a sense-preserving univalent harmonic mapping and is convex in some direction. These results are a generalization of the corresponding recent result of Dorff et al.. The contents of this paper enhance the interest in univalent harmonic mappings, especially when much is not known on the harmonic convolution.