Level sets for functions convex in one direction

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 /. Received by the editors March 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 30C45; Secondary 30C20. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $. 25 per page 442 License or copyright restrictions may apply to redistribution; see http://www.ams. org/journal-terms-of-use LEVEL SETS FOR FUNCTIONS CONVEX IN ONE DIRECTION 443 The proof of this theorem depends on a representation formula for CIA functions and a result relating the total variation of the argument along a curve to the number of times the curve intersects lines through the origin. Lemma A (Royster and Ziegler [8]). A function f belongs to CIA if and only if for some 0 < /x, v < 77 where hv(z) = z/[l — (2cose)z + z2]. A function / analytic in U is said to be starlike in the direction Le: te'e (-oo < / < oo) if the intersection of f(U) with Le is a single segment, half-line, or Le. The following result gives a relation between these functions and functions convex in one direction. Our formulation follows easily from the results in [7]. Lemma B (Robertson [7]). Ifzf'(z) is starlike in the direction Le: te'e (-oo < / < oo) for \z\ < r, thenf(z) is convex in the direction Le + vr/2 for \z\ < r. Finally we need a counting result given in [9] and given as a problem in [2, p. 215]. Lemma C. // <p(z) is analytic in \z\ < r and 0 £ 4>(\z\ = r), then /o2"|Re{^}|^=/;«(^)^, W-r, where n(\L) is the number of times the line te"*1 (-oo < t < oo) intersects the curve H\z\ = r). Although this result is geometrically obvious, for completeness sake we give a proof using the Banach indicatrix. Proof of Lemma C. By replacing <f>(z) by <b(e'az), if necessary, we may assume <b(r) > 0. Let 0 = 6X < 82 < ■ ■ ■ < 6n_1 < On = 277 be all those angles such that <b(re'9k) > 0 and Arg<j>(re'e) varies continuously from 0 to 2ir for 6k < 0 < 6k+1 for each 1 < k < n — 1. Note that if the curve <>(|z| = r) winds around the origin more than once, then « > 3. For each interval Ok < 6 < Ok+l, 1 < k < n — 1, let Hk(0) = Arg<p(re'e). Hence 0 < Hk(8) < 277. Let mk(\p) be equal to the number of roots of the equation Hk(6) = ^, 0 < t¿> < 2t7. The function mk is the Banach indicatrix of the function Hk and thus ¡2mmk(^)d^= [°k+l \dHk(d)\, l<fc«»-l Jo Jek (see Natanson [6, p. 225] for example). Now since \dHk(0)\ = \Re{z(b'(z)/<b(z)}\d6, it follows that rlir , . , r2ir / m{*)d*= / '0 •'0 m'M*! <b(z) de, License or copyright restrictions may apply to redistribution; see http://www.ams. org/journal-terms-of-use