Properties of analytic solutions of three similar differential equations of the second order

Abstract

An analytic univalent in ${\mathbb D}=\{z:\;|z|<1\}$ function $f(z)$ is said to be convex if $f({\mathbb D})$ is a convex domain. It is well known that the condition $\text{Re}\,\{1+zf''(z)/f'(z)\}>0$, $z\in{\mathbb D}$, 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 $\text{Re}\,(f'(z)/\Phi'(z))>0$, $z\in{\mathbb D}$. 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 $0<R\le+\infty$, ${\mathbb D}_R=\{z:\;|z|<R\}$ and $l$ be a positive continuous function on $[0,R)$, which satisfies $ (R-r)l(r)>C,$ $C=\text{const}>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.