Differential inequalities for spirallike and strongly starlike functions

Abstract

Abstract 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 $p(0)=1$ to satisfy Re { e i β p ( z ) } > γ $\operatorname{Re}\{ {\mathrm{e}}^{{\mathrm{i}}\beta } p(z) \} > \gamma $ or | arg { p ( z ) − γ } | < δ $| \arg \{p(z)-\gamma \} |<\delta $ for all z ∈ D $z\in \mathbb{D}$ , where β ∈ ( − π / 2 , π / 2 ) $\beta \in (-\pi /2,\pi /2)$ , γ ∈ [ 0 , cos β ) $\gamma \in [0,\cos \beta )$ , δ ∈ ( 0 , 1 ] $\delta \in (0,1]$ and D : = { z ∈ C : | z | < 1 } $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1 \}$ . The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in D $\mathbb{D}$ .

Keywords

• Carathéodory functions
• Differential subordination
• Starlike functions
• Spirallike functions
• Strongly starlike functions