Analytic Functions Star-Like in One Direction

Robertson, M. S.

doi:10.2307/2370963

Cited by 103 publications

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“…The author [5,6] has shown that if the coefficients are all real and if f(z) is univalent and convex only in the direction of the imaginary axis for \z\ < 1, then again \a n \ ^ 1, but that if the coefficients are complex the results \a n \ ^ n is sharp. For the class of functions f(z) which are close-toconvex in \z\ < 1, the inequalities \a n \ ^n again hold [4].…”

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confidence: 99%

Applications of the subordination principle to univalent functions

Robertson¹

1961

Pacific J. Math.

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confidence: 99%

Applications of the subordination principle to univalent functions

Robertson¹

1961

Pacific J. Math.

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“…It follows that for each r, 0 < r < 1, Re <b(rem) is decreasing for 0 < 0 < m and increasing for tt < 0 < 2tt. Hence, according to a theorem due to Robertson [7], each <j> E N2 maps the unit disc onto a domain that is convex in the direction of the imaginary axis. Moreover, each <p E A2 is real for -1 < z < 1.…”

Section: ] ([Z|(-oo -1] U [1 Oo)])mentioning

confidence: 99%

The Radii of Starlikeness and Convexity of Certain Nevanlinna Analytic Functions

Reade¹,

Todorov²

1981

Proceedings of the American Mathematical Society

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“…The following result gives a relation between these functions and functions convex in one direction. Our formulation follows easily from the results in [7].…”

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“…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.…”

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Level sets for Functions Convex in one Direction

Brown¹

1987

Proceedings of the American Mathematical Society

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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 /.

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Analytic Functions Star-Like in One Direction

Cited by 103 publications

References 0 publications

Applications of the subordination principle to univalent functions

Applications of the subordination principle to univalent functions

The Radii of Starlikeness and Convexity of Certain Nevanlinna Analytic Functions

Level sets for Functions Convex in one Direction

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