### Properties 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  function $f$ is said to be convex if $f\left(\mathbb{D}\right)$ is a convex domain. It is well known that the condition

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  $\left(z\in \mathbb{D}\right)$. 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 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

close-to-convex or convex in $\mathbb{D}$ together with all its derivatives . The results depend on equality or inequality to zero of the parameter ${\gamma }_{2}$.

For example, it is proved that if , ${\gamma }_{2}\ne 0$,
${\gamma }_{0}=p{\beta }_{0}+{\gamma }_{1}={\beta }_{1}+{\gamma }_{2}={\alpha }_{1}{\gamma }_{2}+p{\beta }_{0}{\alpha }_{1}=0$
holds, this equation has a polynomial solution

where the coefficients ${f}_{n}$ are defined by the equality

such that:
1) if and $11\left(p-2\right)\left|{\beta }_{0}\right|}{4}\le 3-\left|{\beta }_{1}\right|$ 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 $\left(41p-50\right)\left|{\beta }_{0}\right|}{8}+4\left|{\alpha }_{0}\right|\le 2-\left|{\beta }_{1}\right|$ and $33\left(p-2\right)\left|{\beta }_{0}\right|}{8}\le 3-\left|{\beta }_{1}\right|$ 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 ${\gamma }_{2}=0$.

PDF (English)

#### Посилання

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DOI: http://dx.doi.org/10.30970/vmm.2021.91.072-086

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