Differential inequalities for spirallike and strongly starlike functions

Abstract

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