The general Brannan coefficient conjecture and Watson's lemma

The coefficients $A_n(\alpha,\beta,\omega)$ in the Maclaurin expansion $(1+\omega z)^{\alpha}(1-z)^{-\beta}= \sum_{n=0}^{\infty} A_n(\alpha,\beta,\omega)z^n$ are studied, where $\omega,z \in \mathbb{C}$ with $|z| < |\omega|=1$, and $\alpha,\beta \in (0,1]$. In 1973 Brannan conjectured that $|A_n(\alpha,\beta,\omega)|\le A_n(\alpha,\beta,1)$ for each positive odd integer $n$, and showed it is true for $n=3$. This has recently been proven for all odd integers $n\ge5$ 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 $\alpha, \beta \in (0,1]$, $0 \le |\arg(\omega)| \le \pi-\phi_0$ and $n=5,7,9,\ldots$, where $\phi_0=0.061$.