Sufficient conditions for starlikeness associated with parabolic region (Q1854154)

The class \(S^*(\alpha)\) of starlike functions of order \(\alpha\) is distinguished by the condition \(\text{Re} (zf'(z)/f(z)) >\alpha\). Analogously, the condition \[ \text{Re}\left\{{zf'(z)\over f(z)}\right\} >\left |{zf'(z)\over f(z)}-1 \right| \] defines the class \(S_p\). In the paper, it is shown that some differential subordinations of an analytic function \(f\) lead to \(f\in S^*(\alpha)\) (or to \(f\in S_p)\).