# Differential inequalities for spirallike and strongly starlike functions

Authors
• 1 Pukyong National University, Busan, 48513, Korea , Busan (South Korea)
• 2 Kyungsung University, Busan, 48434, Korea , Busan (South Korea)
Type
Published Article
Journal
Publisher
Springer International Publishing
Publication Date
Nov 29, 2021
Volume
2021
Issue
1
Identifiers
DOI: 10.1186/s13662-021-03670-9
Source
Springer Nature
Keywords
Disciplines
• Difference Equations, Special Functions and Orthogonal Polynomials
In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that p(0)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(0)=1$\end{document} to satisfy Re{eiβp(z)}>γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{Re}\{ {\mathrm{e}}^{{\mathrm{i}}\beta } p(z) \} > \gamma$\end{document} or |arg{p(z)−γ}|<δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$| \arg \{p(z)-\gamma \} |<\delta$\end{document} for all z∈D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$z\in \mathbb{D}$\end{document}, where β∈(−π/2,π/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta \in (-\pi /2,\pi /2)$\end{document}, γ∈[0,cosβ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma \in [0,\cos \beta )$\end{document}, δ∈(0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta \in (0,1]$\end{document} and D:={z∈C:|z|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{D}:=\{z\in \mathbb{C}:|z|<1 \}$\end{document}. The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{D}$\end{document}.