A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

Wirths, Karl-Joachim

doi:10.4064/ap83-1-10

Cited by 7 publications

(4 citation statements)

References 6 publications

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“…The case n = 3 of (4) was proved in [2] and the cases n = 4 and n = 5 in [9]. According to the above observation this implies that (1) has been proved for n = 2, 3, 4, 5.…”

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confidence: 67%

Sharp inequalities for the coefficients of concave schlicht functions

Avkhadiev¹,

Pommerenke²,

Wirths³

2006

Comment. Math. Helv.

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Abstract. Let D denote the open unit disc and let f : D → C be holomorphic and injective in D. We further assume that f (D) is unbounded and C \ f (D) is a convex domain. In this article, we consider the Taylor coefficients a n (f ) of the normalized expansionand we impose on such functions f the second normalization f (1) = ∞. We call these functions concave schlicht functions, as the image of D is a concave domain. We prove that the sharp inequalitiesare valid. This settles a conjecture formulated in [2].

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“…The case n = 3 of (4) was proved in [2] and the cases n = 4 and n = 5 in [9]. According to the above observation this implies that (1) has been proved for n = 2, 3, 4, 5.…”

mentioning

confidence: 67%

Sharp inequalities for the coefficients of concave schlicht functions

Avkhadiev¹,

Pommerenke²,

Wirths³

2006

Comment. Math. Helv.

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“…Finally in [4] Avkhadiev and Wirths proved (4) in full generality, settling two conjectures of [5] and [6], respectively. In [1], it was also observed that every point of the domain described by (4) has the relation…”

Section: A Point On the Boundary Of This Domain Is Attained If And Onmentioning

confidence: 93%

“…This family of functions is called the family of concave univalent functions with pole at p. The class Co(p) was studied extensively in the recent years. In [1], the third author of the present article proved the following characterization for functions in Co(p) that led to a number of investigations on concave functions.…”

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confidence: 99%

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Concave functions, blaschke products, and polygonal mappings

2009

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We consider the class Co(p) of all conformal maps of the unit disk onto the exterior of a bounded convex set. We prove that the triangle mappings, i.e., the functions that map the unit disk onto the exterior of a triangle, are among the extreme points of the closed convex hull of Co(p). Moreover, we prove a conjecture on the closed convex hull of Co(p) for all p ∈ (0, 1) which had partially been proved by the authors for some values of p ∈ (0, 1).

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On some results of A. E. Livingston and coefficient problems for concave univalent functions

Bhowmik

2010

Arch. Math.

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We consider functions that map the open unit disc conformally onto the complement of a bounded convex set. We call these functions concave univalent functions. In 1994, Livingston presented a characterization for these functions. In this paper, we observe that there is a minor flaw with this characterization. We obtain certain sharp estimates and the exact set of variability involving Laurent and Taylor coefficients for concave functions. We also present the exact set of variability of the linear combination of certain successive Taylor coefficients of concave functions. Mathematics Subject Classification (2000). Primary 30C45; Secondary 30C50.

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A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

Abstract: Abstract. Let D denote the open unit disc and f : D → C be meromorphic and injective in D. We further assume that f has a simple pole at the point p ∈ (0, 1) and an expansion

Cited by 7 publications

References 6 publications

Sharp inequalities for the coefficients of concave schlicht functions

Sharp inequalities for the coefficients of concave schlicht functions

Concave functions, blaschke products, and polygonal mappings

On some results of A. E. Livingston and coefficient problems for concave univalent functions

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