A proof of a conjecture on the convolution of harmonic mappings and some related problems

Abstract

Recently, Kumar, et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with a vertical strip mapping. They verified this conjecture for n = 1, 2, 3 and 4. Moreover, it was proved only for ss =./2. By using of a new method, we settle this conjecture in the affirmative way for all n 2 N and ss 2 (0,.). Moreover, we apply this method to prove some results on the convolutions of harmonic mappings. The proposed new method simplifies calculations and remarkably shortens the proof of the results.