@article{Sheremeta_Trukhan_2021, title={Properties of analytic solutions of three similar differential equations of the second order}, volume={13}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/4528}, DOI={10.15330/cmp.13.2.413-425}, abstractNote={<p>An analytic univalent in ${\mathbb D}=\{z:\;|z|&lt;1\}$ function $f(z)$ is said to be convex if $f({\mathbb D})$ is a convex domain. It is well known that the condition $\text{Re}\,\{1+zf’’(z)/f’(z)\}&gt;0$, $z\in{\mathbb D}$, is necessary and sufficient for the convexity of $f$. The function $f$ is said to be close-to-convex in ${\mathbb D}$ if there exists a convex in ${\mathbb D}$ function $\Phi$ such that $\text{Re}\,(f’(z)/\Phi’(z))&gt;0$, $z\in{\mathbb D}$. S.M. Shah indicated conditions on real parameters $\beta_0,$ $\beta_1,$ $\gamma_0,$ $\gamma_1,$ $\gamma_2$ of the differential equation $z^2w’’+(\beta_0 z^2+\beta_1 z)w’+(\gamma_0z^2+\gamma_1 z+\gamma_2) w=0, $ under which there exists an entire transcendental solution $f$ such that $f$ and all its derivatives are close-to-convex in ${\mathbb D}$. Let $0&lt;R\le+\infty$, ${\mathbb D}_R=\{z:\;|z|&lt;R\}$ and $l$ be a positive continuous function on $[0,R)$, which satisfies $ (R-r)l(r)&gt;C,$ $C=\text{const}&gt;1. $ An analytic in ${\mathbb D}_R$ function $f$ is said to be of bounded $l$-index if there exists $N\in {\mathbb Z}_+$ such that for all $n\in {\mathbb Z}_+$ and $z\in {\mathbb D}_R$ \[\frac{|f^{(n)}(z)|}{n!l^n(|z|)}\le \max\bigg\{\frac{|f^{(k)}(z)|}{k!l^k(|z|)}:\;0\le k\le N\bigg\}.\] Here we investigate close-to-convexity and the boundedness of the $l$-index for analytic in ${\mathbb D}$ solutions of three analogues of Shah differential equation: $z(z-1) w’’+\beta z w’+\gamma w=0$, $(z-1)^2 w’’+\beta z w’+\gamma w=0$ and $(1-z)^3 w’’+\beta(1- z) w’+\gamma w=0$. Despite the similarity of these equations, their solutions have different properties.</p>}, number={2}, journal={Carpathian Mathematical Publications}, author={Sheremeta, M.M. and Trukhan, Yu.S.}, year={2021}, month={Aug.}, pages={413–425} }