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.