On a Coefficient Conjecture of Brannan

Barnard, Roger W.; Pearce, Kent; Wheeler, William

doi:10.1080/17476939708815011

Cited by 15 publications

(30 citation statements)

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“…The case β = 1, α ∈ (0, 1) is still open. Regarding this case partial results were obtained in [3], [4], [6], [7]. We will prove some partial results regarding the case β = 1, and α ∈ (0, 1).…”

Section: Introductionmentioning

confidence: 85%

On Brannan’s Conjecture

Szász¹

2020

Mediterr. J. Math.

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Section: Introductionmentioning

confidence: 85%

On Brannan’s Conjecture

Szász¹

2020

Mediterr. J. Math.

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“…This conjecture, for the special case β = 1, has been established by Brannan [4] (n = 3), J. G. Milcetich [7] (n = 5) and recently by R. W. Barnard, K. Pearce and W. Wheeler [3] (n = 7). For the cases α = β it has been verified by H. Silverman and E. Silvia [8] (n = 3; for the context of their work compare Section 3) and very recently by R. Geisler [6] for n ≤ 33, making heavy use of computer algebra.…”

Section: Introductionmentioning

confidence: 93%

On Brannan's Coefficient Conjecture and Applications

Ruscheweyh

Salinas

2007

Glasgow Math. J.

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Abstract. D. Brannan's conjecture says that for 0 < α, β ≤ 1, |x| = 1, and n ∈ ‫ގ‬ one has |A 2n−1 (α, β, x)| ≤ |A 2n−1 (α, β, 1)|, whereWe prove this for the case α = β, and also prove a differentiated version of the Brannan conjecture. This has applications to estimates for Gegenbauer polynomials and also to coefficient estimates for univalent functions in the unit disk that are 'starlike with respect to a boundary point'. The latter application has previously been conjectured by H. Silverman and E. Silvia. The proofs make use of various properties of the Gauss hypergeometric function.2000 Mathematics Subject Classification. Primary 30C50; Secondary 33C05, 33C45.

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“…Actually, this system is called "complex analysis". The development of complex numbers as an extension of rigorous proofs has provided new and expanding perspectives in many areas of mathematics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Section: History Of Complex Numbersmentioning

confidence: 99%

Complex number and its discovery history

Chen¹,

Zhang²,

Zou³

2023

HSET

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The operations of complex numbers are the main subject of the mathematical analysis area known as complex analysis. It is also known as the theory of functions of a complex variable. The primary research topic in the field of complex analysis is holomorphic functions. These functions are defined on the complex plane, have differentiable properties, and allow for negative values. The residue theorem, the Cauchy integral formula, the Laurent series expansion, etc. are a few concepts, ideas, and methods that are commonly used in research. In particular, over the years, complex analysis in mathematics, physics, and engineering has been extensively used in algebraic geometry, fluid dynamics, quantum mechanics, and other related areas. Two Italian mathematicians, Girolamo Cardano and Raphael Bombelli, discovered complex numbers in the 16th century while attempting to solve a particular algebra, and Cauchy and Riemann extended it in the 19th century. This essay begin with investigation of arithmetic propertity of comlex numbers and then fully explores history development of complex numbers.

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On a Coefficient Conjecture of Brannan

Cited by 15 publications

References 5 publications

On Brannan’s Conjecture

On Brannan’s Conjecture

On Brannan's Coefficient Conjecture and Applications

Complex number and its discovery history

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