Integral transforms of neighborhoods of univalent functions (Q1902773)

Let \(f(z)= z+ \sum^\infty_{k= 2} a_k z^k\) belong to the class \(S\) of normalized univalent functions. Following Ruscheweyh an analytic function \(g(z)= z+ \sum^\infty_{k= 2} b_k z^k\) \((|z|< 1)\) is called to be in the neighborhood \(N_\delta(f)\) if \(\sum^\infty_{k= 2} k|b_k- a_k|\leq \delta\). The author now considers the integral transforms \(P_\lambda\), \(Q_\mu\) \((\lambda, \mu\in \mathbb{C})\) defined by \(P_\lambda[g](z)= \int^z_0 g'(t)^\lambda dt\), \(Q_\mu[g](z)= \int^z_0 (g(t)/t)^\mu dt\) and investigates the problem under which conditions on \(f\) and \(\lambda\), \(\mu\) there is some neighborhood \(N_\delta(f)\) that is mapped into \(S\) by \(P_\lambda\), \(Q_\mu\), respectively. His main result is the following: If \(f\in S\) satisfies \(|f'(z)|\geq m> 0\) and if \(|\lambda|< 1/4\) and \(\delta\leq m(1- \exp(4- 1/|\lambda|))\), then \(P_\lambda[N_\delta(f)]\subset S\). Similarly if \(f\in S\), \(|\mu|< 1/4\) and \(\delta\leq 1- \exp(4- 1/|\mu|)\), then again \(Q_\mu[N_\delta(f)]\subset S\). In case that \(f\) is convex or starlike larger values of \(|\lambda|\), \(|\mu|\), \(\delta\) are allowed. The author's main tool is the Ahlfors univalence criterion \[ ((1- |z|^2) zF''(z)/F(z)- c|z|^2)\leq 1 \] \((|c|\leq 1,\;c\neq 1)\). In addition, well-known distortion theorems are applied.