The general Brannan coefficient conjecture II: Meijer-function approximations

The coefficients $A_n(α,β,ω)$ in the Maclaurin expansion $(1+ωz)^α(1-z)^{-β}=\sum_{n=0}^{\infty} A_n(α,β,ω)z^n$ are considered for $|ω|=1$ and $α,β\in(0,1]$. D. A. Brannan conjectured in a 1973 paper that $|A_n(α,β,ω)|\le A_n(α,β,1)$ for every positive odd integer $n$. The present author recently es