The general Brannan coefficient conjecture and Watson's lemma

The coefficients $A_n(\alpha ,\beta ,\omega )$ in the Maclaurin expansion $(1+\omega z{)}^{\alpha }(1-z{)}^{-\beta }={\sum }_{n=0}^{\mathrm{\infty }}{A}_n(\alpha ,\beta ,\omega )z^n$ are studied, where $\omega ,z\in \mathbb{C}$ with $\mathrm{\mid }z\mathrm{\mid }<\mathrm{\mid }\omega \mathrm{\mid }=1$, and $\alpha ,\beta \in (0,1]$. In 1973 Brannan conjectured that $\mathrm{\mid }{A}_n(\alpha ,\beta ,\omega )\mathrm{\mid }\le A_n(\alpha ,\beta ,1)$ for each positive odd integer $n$, and showed it is true for $n=3$. This has recently been proven for all odd integers $n\ge 5$ by a number of authors in aggregate for the special case $\beta =1$. 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 $\alpha ,\beta \in (0,1]$, $0\le \mathrm{\mid }\arg (\omega )\mathrm{\mid }\le \pi -{\phi }_0$ and $n=5,7,9,\dots$, where ${\phi }_0=0.061$.