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Selected Publications
On extreme points of subordination families with a convex majorant, Journal of Mathematical Analysis and Applications, vol. 145, no. 1, (1990), 216 - 231
Abstract. Let B0 denote the set of functions φ analytic in the unit disc and satisfying |φ (z) | < 1, φ (0) = 0. Suppose F is analytic in the unit disc and maps it onto a convex domain D other than a strip, a wedge or a half plane. Let s(F) denote the subordination family of F and Es(F) the set of its extreme points. Hallenbeck and MacGregor asked in their monograph [Linear problems and convexity techniques in Geometric Function Theory, Pitman, New York, 1984, page 144] whether the set Es(F) ever lies between the minimal ({Foφ | φ ∈ B0 , limr --> 1|φ (reiθ) | = 1 a.e.}) and the maximal ({Foφ | φ is an extreme point of B0}) sets. We prove that this always happens when the boundary of D contains a line segment.
Extreme points of classes of analytic functions with positive real part and restricted second coefficient, Math. Japonica 36, no. 2 (1991), 221 - 223
On properties of extreme points of subordination families with a convex majorant, Math. Japonica 41, no. 3 (1995), 537 - 543
Abstract. Let Δ be the open unit disc in the plane. Suppose F is analytic in Δ and maps it onto a convex domain D. Let s(F) denote the subordination family of F and Es(F) the set of its extreme points. For every f ∈ s(F), let f(θ) = limr-->1f(reiθ) be the boundary function of f, let {f(θ)} denote the set of all existing radial limits of f, and define λ(θ) = dist(f(θ), ∂F(Δ)) almost everywhere on ∂Δ. We show two properties of extreme points of s(F). In the case when D is the interior of a convex polygon, we prove that if f ∈ Es(F) then the closure of {f(θ)} must contain ∂D. We also prove that if D is any convex domain and the boundary function f(θ) of an extreme point of s(F) is continuous then there exists a point e ∈ E∂D such that for every neighborhood N(e) of e the integral of log λ(θ) over { θ ∈ ∂Δ: f(θ) ∈ N(e)} is -∞.
The span and semispan of some simple closed curves, Proceedings of the American Mathematical Society, vol. 111, no. 1 (1991)
Abstract. The spans of simple closed curves that are boundaries of convex regions are determined. The following question of Lelek is answered affirmatively for these curves: Are the span and the semispan of a simple closed curve equal?
On the span of simple closed curves, Houston Journal of Mathematics, vol. 20, no. 3 (1994)
Abstract. We show that the span of a simple closed curve is not smaller than the span of the boundary of any convex region inscribed in it. The span and semispan of some types of simple closed curves are determined and proved to be equal.
Estimates of spans of a simple closed curve involving mesh, Houston Journal of Mathematics, vol. 26, no. 4 (2000)
http://hjm.math.uzh.ch/pdf26%284%29/07hallenbeck.pdf
Abstract. We show that the dual effectively monotone span of a simple closed curve X in the plane does not exceed the infimum of the set of positive numbers m such that a chain with mesh m covers X. We also include a short direct proof of a known inequality σ(X) ≤ ε(X).
On the span of starlike curves, Scientiae Mathematicae Japonicae, 55, no. 3 (2002), 547-554, e5, 507-514
Abstract. We prove that the dual monotone span of a starlike curve X is not smaller than the infimum ε(X) of the set of positive numbers m such that a chain with mesh m covers X.
Span mates and mesh, Glasnik Matematicki, vol. 37(57), (2002), 175-186
http://web.math.hr/glasnik/37.1/37%281%29-15.pdf
Abstract. We identify a class of starlike curves with the span and the semispan equal to the infimum of the set of meshes of the covering chains. The same result is obtained for a class of indented circles as defined by West [Spans of simple closed curves, Glasnik Matematicki 24(44), (1989), 405-415].
Span mates and span, Scientiae Mathematicae Japonicae 59, no. 1 (2004), 79-83, e9, 119-123
Abstract. We show that the span mate width of a simple closed curve does not exceed the span of that simple closed curve.
A note on Cook's inequality for simple closed curves, Scientiae Mathematicae Japonicae, 69, No. 2 (2009), 277-279 : e2009, 79-81
Abstract. The question of Howard Cook: "Do there exist, in the plane, two simple closed curves Y and X, such that X is in the bounded complementary domain of Y, and the span of X is greater than the span of Y?" is answered, in the negative, for several types of simple closed curve pairs.
The Abu-Muhanna conjecture on support points, Scientiae Mathematicae Japonicae, 71, No. 3 (2010), 327-330 : e-2010, 129-132
Abstract. We determine support points of the subordination family of an arbitrary bounded analytic function, thereby proving the Abu-Muhanna conjecture [J. London Math. Soc. (2), 29 (19840].
A theorem on the subject of Cook's inequality, Scientiae Mathematicae Japonicae, 72, No. 2 (2010), 111-116 : e2010, 387-392
Abstract. We show that the span of an arbitrary simple closed curveX does not exceed the span of any starlike curve contained in the clousure of the unbounded component of the complement of X.
With D. J. Hallenbeck:
Extreme points and support points of subordination families with p-valent majorants, Annales Polonici Mathematici L (1989), 93 - 115
Abstract. Let Sp*, Kp and Cp denote the p-valent starlike, convex and close-to-convex analytic functions in the unit disc normalized so as to have p zeros at the origin. We study the classes defined by subordination to Sp*, Kp and Cp. We determine the support points of these classes and the extreme points of their closed convex hulls. This information is used to solve extremal problems.
Extreme points and support points of subordination families, Journal of Mathematical Analysis and Applications 251, 157-166 (2000)
Abstract. This paper surveys the main results and open questions in the theory of extreme points and support points of subordination families of analytic functions.
Classes of analytic functions subordinate to convex functions and extreme points, J. Math. Anal. Appl. 282 (2003), 792 - 800
Abstract. In this paper we prove two sufficient conditions for an analytic function f to be an extreme point of the set of functions subordinate to a given convex mapping F when the image of the unit disk under F is a convex domain other than a half-plane, a strip or an infinite wedge.
With David P. Bellamy:
Extreme points of some classes of analytic functions with positive real part and a prescribed set of coefficients, Complex Variables, vol. 17 (1991), 49 - 55
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