On the Fekete-Szegö theorem for close-to-convex functions. (English) Zbl 0868.30015

Let \(f(z)= z+a_2z^2+a_3z^3\cdots\) be holomorphic in the unit disc such that there exists a function \(g(z)=z+\cdots\) holomorphic in the unit disk with \(\text{Re }{zg'(z)\over g(z)}>\beta\) for some fixed \(\beta\in[0,1)\), \(|z|<1\) and \(\text{Re }{zf'(z)\over g(z)}>\alpha\) for some fixed \(\alpha\in[0,1)\), \(|z|<1\). For such functions the authors find sharp bounds for \(|a_3-\mu a^2_2|\), \(\mu\) real.
Further references on similar problems are: W. Koepf [Proc. Am. Math. Soc. 101, 89-95 (1987; Zbl 0635.30019), Arch. Math. 49, 420-433 (1987; Zbl 0635.30020)].


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)