On meromorphically starlike functions of order α and type β, which satisfy Shah’s differential equation - Trukhan, Yu.S.; Mulyava, O.M. (Carpathian Mathematical Publications, 2018)
According to M.L. Mogra, T.R. Reddy and O.P. Juneja an analytic in D_0={z: 0<|z|<1} function f(z)=1z+_n=1^∞f_n z^n is said to be meromorphically starlike of order α[0, 1) and type β(0, 1] if |zf'(z)+f(z)|<β|zf'(z)+(2α-1)f(z)|, zD_0. Here we investigate conditions on complex parameters β_0, β_1, γ_0, γ_1, γ_2, under which the differential equation of S. Shah z^2 w''+(β_0 z^2+β_1 z) w'+(γ_0 z^2+γ_1 z+γ_2)w=0 has meromorphically starlike solutions of order α[0, 1) and type β(0, 1]. Beside the main case n+γ_2 ot=0, n1, cases γ_2=-1 and γ_2=-2 are considered. Also the possibility of the existence of the solutions of the form f(z)=1z+_n=1^mf_n z^n, m2, is studied. In addition we call an analytic in D_0 function f(z)=1z+_n=1^∞f_n z^n meromorphically convex of order α[0,1) and type β(0,1] if |zf''(z)+2f'(z)|<β|zf''(z)+2α f'(z)|, zD_0 and investigate sufficient conditions on parameters β_0, β_1, γ_0, γ_1, γ_2 under which the differential equation of S. Shah has meromorphically convex solutions of order α[0, 1) and type β(0, 1]. The same cases as for the meromorphically starlike solutions are considered.