Harmonic mappings and its directional convexity
For any μ _j (μ _jC,μ _j =1,j=1,2), we consider the rotations f_μ _1 and F_μ _2 of right half-plane harmonic mappings f,FS_H which are CHD with the prescribed dilatations ω _f(z)=( a-z) /( 1-az) for some a ( -1<a<1) and ω _F(z)= e^iθ z^n ( nN,θ R) , ω _F(z)= ( b-z) /( 1-bz) , ω _F(z)=( b-ze^iφ ) /( 1-bze^iφ ) % (-1<b<1,φ R), respectively. It is proved that the convolution f_μ _1 * F_μ _2 S_H and is convex in the direction of μ _1μ _2 under certain conditions on the parameters involved. Mathematics Subject Classification (2010): 31A05, 30C45, 30C55. , number=4, journal=Studia Universitatis Babeș-Bolyai Mathematica, author=SHARMA, Poonam and Omendra MISHRA, Omendra MISHRA, year=2021, month=Dec., pages=677–690