Properties of polynomial solutions of a differential equation of the second order with polynomial coefficients of the second degree

Abstract

An analytic univalent inis necessary and sucient for the convexity of f . Function f is said to be closeto-convex if there exists a convex inClose-to-convex function f has a characteristic property that the complement G of the domain f (D) can be lled with rays which start from ∂G and lie in G. Every close-to-convex in D function f is univalent in D and, therefore, f (0) = 0.We indicate conditions on parameters β0, β1, γ0, γ1, γ2 and α0, α1, α2 of the dierential equationunder which this equation has a polynomial solutionclose-to-convex or convex in D together with all its derivatives f (j) (1 ≤ j ≤ p − 1). The results depend on equality or inequality to zero of the parameter γ2.For example, it is proved that if p ≥ 3, γ2 = 0, γ0 = pβ0 + γ1 = β1 + γ2 = α1γ2 + pβ0α2 = 0