A subordination theorem for spirallike functions (Q1590695)

Let \(G(\lambda)\) denote the class of functions \(f(z)= z+\sum^\infty_{n=2} a_nz^n\), analytic in the unit disk \(E\) and satisfying \[ \sum^\infty_{n=2} [1+(n- 1)\text{sec }\lambda]|a_n|\leq 1,\;\Biggl(|\lambda|< {\pi\over 2}\Biggr). \] Theorem: Let \(f\in G(\lambda)\), \(g\) is analytic in \(E\), \(g(0)= g'(0)- 1= 0\) and \(g(E)\) is convex, \(f*g\) means Hadamard product. Then \({1+ \text{sec }\lambda\over 2(2+\text{sec }\lambda)}(f* g)(z)\) is subordinate to \(g(z)\) in \(E\). The factor \({1+ \text{sec }\lambda\over 2(2+ \text{sec }\lambda)}\) is best possible.