Properties 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. It is well known that the condition
$\mathrm{Re} \left\{1+\raisebox{1ex}{$\mathrm{zf}\text{'}\text{'}\left(\mathrm{z}\right)$}\!\left/ \!\raisebox{-1ex}{$\mathrm{f}\text{'}\left(\mathrm{z}\right)$}\right.\right\}>0 \mathrm{z}\in \mathbb{D}$
is necessary and sufficient for the convexity of $f$. Function $f$ is said to be close-to-convex if there exists a convex in $\mathbb{D}$ function $\Pi$ such that $\mathrm{Re} \left(\raisebox{1ex}{$\mathrm{f}\text{'}\left(\mathrm{z}\right)$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\Pi }\text{'}\left(\mathrm{z}\right)$}\right.\right)>0$ $\left(z\in \mathbb{D}\right)$. Close-to-convex function $f$ has a characteristic property that the complement $G$ of the domain $f\left(\mathbb{D}\right)$ can be filled with rays which start from $\partial \mathbb{D}$ and lie in $\mathbb{D}$. Every close-to-convex in $\mathbb{D}$ function $f$ is univalent in $\mathbb{D}$ and, therefore, $f\text{'}\left(0\right)\ne 0$.
We indicate conditions on parameters ${\beta }_0, {\beta }_1, {\gamma }_0,{\gamma }_1,{\gamma }_2$ and ${\alpha }_0,{\alpha }_1,{\alpha }_2$ of the differential equation
$z^{2}w"+\left({\beta }_0+{\beta }_{1}z\right)w\text{'}+\left({\gamma }_0{z}^2+{\gamma }_{1}z+{\gamma }_2\right)w={\alpha }_0{z}^2+{\alpha }_{1}z+{\alpha }_{2,}$
under which this equation has a polynomial solution
$f\left(z\right)=\sum _{n=0}^p{f}_n{z}^n (\mathrm{deg} \mathrm{f}=\mathrm{p}\ge 2)$
close-to-convex or convex in $\mathbb{D}$ together with all its derivatives $f^{\left(j\right)} (1\le j\le p-1)$. The results depend on equality or inequality to zero of the parameter ${\gamma }_2$.

For example, it is proved that if $p\ge 3,$, ${\gamma }_2\ne 0$,
${\gamma }_0=p{\beta }_0+{\gamma }_1={\beta }_1+{\gamma }_2={\alpha }_1{\gamma }_2+p{\beta }_0{\alpha }_1=0$
holds, this equation has a polynomial solution
$f\left(z\right)=\raisebox{1ex}{${\alpha }_2$}\!\left/ \!\raisebox{-1ex}{${\gamma }_2$}\right.+z+\frac{{\alpha }_0+(p-1){\beta }_0}{2+{\beta }_1} z^2+\sum _{n=3}^p{f}_n{z}^n,$
where the coefficients $f_n$ are defined by the equality
$f_n=\frac{(p-n+1){\beta }_0}{(n-1)(n+{\beta }_1)} f_{n-1 (3\le n\le p),}$
such that:
1) if $\raisebox{1ex}{$\left(11p-14\right)\left|{\beta }_0\right|$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\left|{\alpha }_0\right|\le 2-\left|{\beta }_1\right|$ and $\raisebox{1ex}{$11\left(p-2\right)\left|{\beta }_0\right|$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\le 3-\left|{\beta }_1\right|$ then $f$ is close-to-convex in $\mathbb{D}$ together with all its derivatives $f^{\left(j\right)}$ $\left(1\le j\le p-1\right)$;
2) if $\raisebox{1ex}{$\left(41p-50\right)\left|{\beta }_0\right|$}\!\left/ \!\raisebox{-1ex}{$8$}\right.+4\left|{\alpha }_0\right|\le 2-\left|{\beta }_1\right|$ and $\raisebox{1ex}{$33(p-2)\left|{\beta }_0\right|$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\le 3-\left|{\beta }_1\right|$ then $f$ is convex in $\mathbb{D}$ together with all its derivatives $f^{\left(j\right)}$ $\left(1\le j\le p-1\right)$.
A similar result is obtained in the case ${\gamma }_2=0$.