Convolution Properties of Some Harmonic Mappings in the Right Half-Plane

Abstract

It is well known that harmonic convolution of two normalized right half-plane mappings is convex in the direction of the real axis, provided the convolution function is locally univalent and sense-preserving in \(E = \{z: |z|<1\}\). Further, it is also known that the condition of local univalence and sense-preserving in E on the convolution function can be dropped when one of the convoluting functions is the standard right half-plane mapping with dilatation \(\displaystyle -z\) and other is the right half-plane mapping with dilatation \(e^{i\theta } z^n,\,n=1,2,\)\(\theta \in \mathbb {R}.\) This result does not hold for \(n=3,4,5,\ldots .\) In this paper, we generalize this result by taking the dilatation of one of the right half-plane mappings as \(e^{i\theta } z^n\)\((\,n\in \mathbb {N},\theta \in \mathbb {R})\) and that of the other as \(\displaystyle {(a-z)}/{(1-az)},\)\(a\in (-1,1).\) We shall prove that our result holds true for all \(n\in \mathbb {N},\) provided the real constant a is restricted in the interval \(\left[ {(n-2)}/{(n+2)},1\right) \). The range of the real constant a is shown to be sharp.