The Critical Points of a Typically-Real Function

Abstract

Abstract.The critical points of a typically-real function cannot lie too close to the real axis. By adding a mild restriction, we determine Dk the domain of variability of a A:th order critical point. Similar results are obtained for a kth order branch point. We determine the domain of univalence for typically-real functions and propose a reasonable conjecture for the domain of ¿-valence.

Subordinations for typically-real and related functions

“…Theorem 3 verifies a conjecture of Goodman [4]. It proves that for each fe T the region/(G) is convex in the direction of the imaginary axis.…”

“…More generally, we prove the following theorem. [4], [7]). The function v (z) = z/[l + (1 -z 2 ) 1/2 ] maps E onto G. A relationship between Tand C is given by the next result.…”