A Proof of a Conjecture on the Convolution of Harmonic Mappings and Some Related Problems

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 β = π/ 2 . By using of a new method, we settle this conjecture in the affirmative way for all n ε N and β ε (0 , π ) . 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.