The general Brannan coefficient conjecture II: Meijer-function approximations

Abstract

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 considered for $|\omega |=1$ and $\alpha ,\beta \in (0,1]$. D. A. Brannan conjectured in a 1973 paper that $|A_n(\alpha ,\beta ,\omega )|\le A_n(\alpha ,\beta ,1)$ for every positive odd integer $n$. The present author recently established the conjecture outside a small neighbourhood of $\omega =-1$. The remaining range is treated here by combining compound Laplace integral representations with two types of local approximation: a Meijer $G$ function approximation for $n|\arg (-\omega )|$ bounded, and a modified Watson approximation for the complementary range. The resulting lower bounds reduce the problem to numerical positivity checks for explicit functions on compact parameter sets. These computations verify the inequality for all $\alpha ,\beta \in (0,1]$ and all odd integers $n\ge 5$, and hence, together with Brannan's result for $n=3$, complete the proof of his conjecture.

arXiv commentCorrected Lemmas 6.1 and 6.2, allowing the dependence on ζto be eliminated analytically, so no numerical search for critical points in ζis needed. Independent optimization calculations were added to check the numerical infima, together with further checks of the Meijer G-function evaluations