Differential inequalities for spirallike and strongly starlike functions

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^iβ p(z) \ > γ Re { e i β p ( z ) } > γ or | (z)-γ \ |<δ | arg { p ( z ) − γ } | < δ for all z∈ D z ∈ D , where β ∈ (-π /2,π /2) β ∈ ( − π / 2 , π / 2 ) , γ ∈ [0,cos β ) γ ∈ [ 0 , cos β ) , δ ∈ (0,1] δ ∈ ( 0 , 1 ] and D:=∈ 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 D D .