In this article, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete–Szegő problem for normalized concave functions with a pole in the unit disk.
"Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions." Kyoto J. Math. 58 (2) 227 - 241, June 2018. https://doi.org/10.1215/21562261-2017-0015