On subordination of certain subclass of analytic functions (Q1363329)

Let \(P_n [\alpha,M]\) \((\alpha \geq0,\;m>0)\) be the class of functions \(f(z) =z+ \sum^\infty_{k=n+1} a_kz^k\) analytic in \(E=\{z: |z|<1\}\), which satisfy \(|f'(z)+ \alpha zf''(z)- 1|< M(z\in E)\). The method of differential subordination is used to prove the sharp upper bounds for \(|f'(z) |\), \(|f(z) |\) as well as the sharp lower bounds for \(\text{Re} f'(z)\), \(\text{Re} f(z)\). The univalence problem of functions within the class \(P_n (\alpha,M)\) is discussed.