The general Brannan coefficient conjecture I: Watson-lemma approximations

Abstract

The coefficients $A_n(\hat{I}\pm ,{\hat{I}}^2,\ddot{I}‰)$ in the Maclaurin expansion $(1+\ddot{I}‰z{)}^{\hat{I}}\pm (1-z{)}^{-{\hat{I}}^2}={\sum }_{n=0}^{\infty }{A}_n(\hat{I}\pm ,{\hat{I}}^2,\ddot{I}‰)z^n$ are studied, where $\ddot{I}‰,z\in \mathbb{C}$ with $|z|<|\ddot{I}‰|=1$, and $\hat{I}\pm ,{\hat{I}}^2\in (0,1]$. In 1973 Brannan conjectured that $|A_n(\hat{I}\pm ,{\hat{I}}^2,\ddot{I}‰)|\le A_n(\hat{I}\pm ,{\hat{I}}^2,1)$ 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 ${\hat{I}}^2=1$. 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 $\hat{I}\pm ,{\hat{I}}^2\in (0,1]$, $0\le |\arg (\ddot{I}‰)|\le \ddot{I}€-\ddot{I}{•}_0$ and $n=5,7,9,\dots$, where $\ddot{I}{•}_0=0.061$.

v.2: Minor corrections, used CIF in proof of Lemma 2.1, and re-scaled w_n for conciseness v.3: Title and sec. 6 updated to reflect sequel paper which covers phi in [0,phi_0] (https://arxiv.org/abs/2606.11621)