Quantum Approach on Convolution of Harmonic Univalent Mappings

Abstract

In this paper, we construct a new family of locally univalent and sense-preserving harmonic mappings Tc,q [f] by using quantum approach in the open unit disk D. We prove that the convolution of a harmonic convex mapping Tc,q [f] with a right half-plane mapping is q-convex in the direction of the real axis provided that the convolution is locally univalent and sense-preserving. Further, we show that the convolution of Tc,q [f] with a vertical strip mapping is also q-convex in the direction of the real axis. In particular, the results in this paper generalize or improve (in certain cases) the corresponding results obtained by recent researchers.