Properties of analytic solutions of three similar differential equations of the second order - Sheremeta, M.M.; Trukhan, Yu.S. (Carpathian Mathematical Publications, 2021)
An analytic univalent in D={z: |z|<1} function f(z) is said to be convex if f(D) is a convex domain. It is well known that the condition Re {1+zf''(z)/f'(z)}>0, zD, is necessary and sufficient for the convexity of f. The function f is said to be close-to-convex in D if there exists a convex in D function Φ such that Re (f'(z)/Φ'(z))>0, zD. S.M. Shah indicated conditions on real parameters β_0, β_1, γ_0, γ_1, γ_2 of the differential equation z^2w''+(β_0 z^2+β_1 z)w'+(γ_0z^2+γ_1 z+γ_2) w=0, under which there exists an entire transcendental solution f such that f and all its derivatives are close-to-convex in D. Let 0<R+∞, D_R={z: |z|<R} and l be a positive continuous function on [0,R), which satisfies (R-r)l(r)>C, C=const>1. An analytic in D_R function f is said to be of bounded l-index if there exists NZ_+ such that for all nZ_+ and zD_R \[|f^(n)(z)|n!l^n(|z|){|f^(k)(z)|k!l^k(|z|): 0kN}.\] Here we investigate close-to-convexity and the boundedness of the l-index for analytic in D solutions of three analogues of Shah differential equation: z(z-1) w''+β z w'+γ w=0, (z-1)^2 w''+β z w'+γ w=0 and (1-z)^3 w''+β(1- z) w'+γ w=0. Despite the similarity of these equations, their solutions have different properties.