Properties of analytic solutions of three similar differential equations of the second order
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...
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Vasyl Stefanyk Precarpathian National University
2021-08-01
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Series: | Karpatsʹkì Matematičnì Publìkacìï |
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Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/4528 |
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author | M.M. Sheremeta Yu.S. Trukhan |
author_facet | M.M. Sheremeta Yu.S. Trukhan |
author_sort | M.M. Sheremeta |
collection | DOAJ |
description | 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. |
first_indexed | 2024-04-24T08:56:36Z |
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institution | Directory Open Access Journal |
issn | 2075-9827 2313-0210 |
language | English |
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publisher | Vasyl Stefanyk Precarpathian National University |
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spelling | doaj.art-2c014091873f497595dd4db77587be0e2024-04-16T07:07:41ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102021-08-0113241342510.15330/cmp.13.2.413-4253959Properties of analytic solutions of three similar differential equations of the second orderM.M. Sheremeta0Yu.S. Trukhan1Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, UkraineIvan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, UkraineAn 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.https://journals.pnu.edu.ua/index.php/cmp/article/view/4528close-to-convexity$l$-indexdifferential equation |
spellingShingle | M.M. Sheremeta Yu.S. Trukhan Properties of analytic solutions of three similar differential equations of the second order Karpatsʹkì Matematičnì Publìkacìï close-to-convexity $l$-index differential equation |
title | Properties of analytic solutions of three similar differential equations of the second order |
title_full | Properties of analytic solutions of three similar differential equations of the second order |
title_fullStr | Properties of analytic solutions of three similar differential equations of the second order |
title_full_unstemmed | Properties of analytic solutions of three similar differential equations of the second order |
title_short | Properties of analytic solutions of three similar differential equations of the second order |
title_sort | properties of analytic solutions of three similar differential equations of the second order |
topic | close-to-convexity $l$-index differential equation |
url | https://journals.pnu.edu.ua/index.php/cmp/article/view/4528 |
work_keys_str_mv | AT mmsheremeta propertiesofanalyticsolutionsofthreesimilardifferentialequationsofthesecondorder AT yustrukhan propertiesofanalyticsolutionsofthreesimilardifferentialequationsofthesecondorder |