Multivalent functions convex in one direction (Q1100582)

Let \({\mathbb{C}}\) denote the complex plane, D the unit disk in \({\mathbb{C}}\), H(D) the class of functions holomorphic in D, S the usual class of functions normalized and univalent in D, and SR(p) the class of functions ``starlike in the direction of the real axis of order p'' introduced by Hollingsworth and Wright. (Functions in the last class are characterized by a product representation.) A region \(\Omega\subset {\mathbb{C}}\) is called convex in the vertical direction if \(\ell \cap \Omega\) is empty or connected for every vertical line \(\ell\). A univalent function \(f\in H(D)\) is called convex in the vertical direction if f(D) is convex in the vertical direction. The authors introduce three plausible definitions for p-valent functions convex in the vertical direction: \(f\in CV_ A(p)\) if zf'\(\in SR(p)\); \(f\in CV_ B(p)\) if \(f=P\circ \phi\) where P is a polynomial of degree p, \(\phi\in S\), \(\phi\) (D) contains all critical points of P, and for each vertical line \(\ell\) and each component \(\Gamma\) of \(P^{-1}(\ell)\), \(\Gamma\cap \phi (D)\) is empty or connected; and \(f\in CV_ c(p)\) if it satisfies a set of geometric conditions. The main result of this paper is that the three classes are actually identical. A second result is that this class is (essentially) closed under local uniform convergence. The proofs are very interesting and are quite geometric and graph theoretic in nature.