2026/02/17 by T. M. Dunster · 1 citation
Mathematics · #math.CA #math.CV
The coefficients An(α,β,ω) in the Maclaurin expansion (1+ωz)α(1-z)-β= ∑n=0∞ An(α,β,ω)zn are studied, where ω,z ∈ ℂ with |z| < |ω|=1, and α,β∈ (0,1]. In 1973 Brannan conjectured that |An(α,β,ω)|≤ An(α,β,1) for each positive odd integer n, and showed it is true for n=3. This has recently been proven for all odd integers n≥5 by a number of authors in aggregate for the special case β=1. In this paper hypergeometric integral representations and Watson-type approximations are utilised, from which the general problem is reduced to numerically evaluating the minima of certain simple, explicit, slowly-varying functions over compact domains. From the positivity of these constants it is shown that the conjecture holds for α, β∈ (0,1], 0 ≤ |arg(ω)| ≤ π-ϕ0 and n=5,7,9,…, where ϕ0=0.061.