On a refinement of a theorem of Landau on Koebe domains

We state and prove a refinement of a classical theorem due to Landau on the Koebe domains for certain families of holomorphic functions introduced by A. W. Goodman. Our geometric approach in this article enables us to derive several statements of interest, which would not be produced via the methods in Goodman's paper, as immediate corollaries of the proof of the main theorem.