### 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 $\mathbb{D}=\left\{z:\left|z\right|<1\right\}$ 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 . We indicate conditions on real parameters and 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

close-to-convex in $\mathbb{D}$ together with all its derivatives ${f}^{j}$ $\left(1\le j\le p-1\right)$.

PDF (English)

#### Посилання

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

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