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

Анотація

An analytic univalent in $\mathbb{D}=\left\{z:\left|z\right|<1\right\}$ function $f$ is said to be convex if $f\left(\mathbb{D}\right)$ is a convex domain and is said to be close-to-convex if there exists a convex in $\mathbb{D}$ function $\Phi$ such that $\mathit{R}\mathit{e}\left(\raisebox{1ex}{$f\text{'}\left(z\right)$}\!\left/ \!\raisebox{-1ex}{$\Phi \text{'}\left(z\right)$}\right.\right)>0 (z\in \mathbb{D})$. We indicate conditions on real parameters ${\beta }_0, {\beta }_1, {\gamma }_0, {\gamma }_1, {\gamma }_2$ and ${\alpha }_0, {\alpha }_1, {\alpha }_2$ of the differential equation
$z^{2}w\text{'}\text{'}+\left({\beta }_0{z}^2+{\beta }_{1}z\right)w\text{'}+\left({\gamma }_0{z}^2+{\gamma }_{1}z+{\gamma }_2\right)w={\alpha }_0{z}^2+{\alpha }_{1}z+{\alpha }_2,$
under which this equation has a polynomial solution
$f\left(z\right)=\sum _{n=0}^p{f}_n{z}^n \left(\mathit{d}\mathit{e}\mathit{g} f=p\ge 2\right)$
close-to-convex in $\mathbb{D}$ together with all its derivatives $f^j$ $\left(1\le j\le p-1\right)$.