Close-to-convexity of polynomial solutions of a differential equation of the second order with polynomial coefficients of the second degree

Myroslav Sheremeta, Yuriy Trukhan

Анотація


An analytic univalent in D=z:z<1 function f is said to be convex if f(D) is a convex domain and is said to be close-to-convex if there exists a convex in D function Φ such that Ref'(z)Φ'(z)>0 (zD). We indicate conditions on real parameters β0, β1, γ0, γ1, γ2 and α0, α1, α2 of the differential equation
z2w''+β0z2+β1zw'+γ0z2+γ1z+γ2w=α0z2+α1z+α2,
under which this equation has a polynomial solution
f(z)=n=0pfnzn  deg f=p2
close-to-convex in D together with all its derivatives fj 1jp-1.

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DOI: http://dx.doi.org/10.30970/vmm.2020.90.092-104

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