Differential inequalities for spirallike and strongly starlike functions

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 \} |