On Brannan's Coefficient Conjecture and Applications

Abstract

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.

On Brannan’s Conjecture

“…where n is a natural number. Partial results regarding this question were already proved in [1], [2], [5], [8].…”

The solution of the Brannan conjecture

“…where n is a natural number. Partial results regarding this question were already proved in [1], [2], [5], [8]. Concerning the case β = 1, and α ∈ (0, 1) partial results have been proved in [3], [4], [6], [7], [9].…”