Level sets for Functions Convex in one Direction

Abstract

Abstract.Goodman and Saff conjectured that if / is convex in the direction of the imaginary axis then so are the functions \f(rz) for all 0 < r < \2 -1, i.e., the level sets f(\z\ < r) are convex in the direction of the imaginary axis for 0 < r < \¡2 -1. A weak form of this conjecture is proved and a question of Brannan is answered negatively.Let Ur= [z:\z\ < r) and let 5 denote the class of all functions f(z) = z + a2z2 + ■ ■ ■ analytic and univalent in U = Uv For / e S there are several geometric properties possessed by f(U) that are inherited by its level sets Gr = f(Ur) for all 0 < r < 1. For example if f(U) is either convex, starlike, or close-to-con vex, then so are its level sets Gr for all 0 < r < 1.An analytic function / is said to be convex in the direction of a line Le: te'8 (-co < t < oo) if the intersection of f(U) with each line parallel to Le is either a connected set or empty. Let CIA denote those functions / for which f(U) is convex in the direction of the imaginary axis with /(0) = 0 and /'(0) = 1. Since CIA functions are close-to-convex, they are univalent. It came as a bit of a surprise when Hengartner and Schober [4] constructed an example where / e CIA but the corresponding level sets Gr were not convex in the direction of the imaginary axis for all r sufficiently close to 1. A more quantitative result was obtained by Goodman and Saff [3]. They were able to prove that for each ^2 -1 < r < 1 there exists an /e CIA for which |/(rz) £ CIA. Hence they conjectured that if /e CIA then jf(rz) e CIA for all 0 < r < y2 -1, i.e., the level sets Gr are convex in the direction of the imaginary axis for 0 < r < y2 -1.In [1, Problem 6.53] Brannan asked whether or not: If / e CIA and 7-J(rQz) £ CIA for some 0 < r0 < 1, does this imply that jf(rz) £ CIA for all r0 < r < 1? This question was motivated by the example constructed by Hengartner and Schober [4]. In this note we prove a weaker form of the Goodman-Saff conjecture and answer Brannan's question. Our main result is the following theorem.Theorem. /// e CIA then there exists a set I c [0, tr] of positive measure such that jf(rz) is convex in the directions L8: te'6 (-oo < t < oo) for all 0 < r < y2 -1 and all 0 e /.

Convexity of a Set on the Plane in a Given Direction

“…The condition of y-convexity arose in [8,9]. In the works devoted to the univalent function theory [28,29], this condition appears as the convexity in the direction of the imaginary axis. An analogous condition for the direction of an arbitrary line was called the θ-convex linearity in [10], and in [17,[29][30][31], it was called the convexity to direction, which is not successful from our point of view, since by Theorems 6.5 it reduces to the convexity in two directions normal to this line.…”

Koebe sets for the class of functions convex in two directions

“…These classes were discussed, among others, by Hengartner and Schober, Goodman and Saff, Ciozda, Brown, and Prokhorov (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). The set K α , α ∈ (0, 1) has not been discussed yet.…”