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 $n\ge5$, and hence, together with Brannan's result for $n=3$, complete the proof of his conjecture.

While long treated as empirical fits, Hill functions have been postulated to be the universal Hopfield barrier for sharpness of input-output responses by Martinez-Corral, Nam, DePace, and Gunawardena. A Hopfield barrier is a fundamental limit on how well biological systems can process information without expending energy. Their case rested on numerical findings for Hill coefficients $4$ and $6$. We give a precise formulation and proof of this: measuring sharpness by the supremum of the derivative in semi-log scale, any rational function $r(x)=(\alpha_0+\alpha_1 x+ \cdots +\alpha_n x^n)/(\beta_0 + \beta_1 x+ \cdots + \beta_n x^n)$ with real coefficients $0\leq \alpha_i\leq \beta_i$ has sharpness at most $n/4$, with equality if and only if $r$ is a Hill function with Hill coefficient $n$.

Let $\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $\pi$, and let $A_{\psi_n}$ be the Laplacian with word length \[ \psi_n(k) = \min(k,n-k). \] We prove the sharp log-Sobolev inequality \[ \text{Ent}_{\pi}(f^2) \le 2\pi(f A_{\psi_n} f), \qquad f:\mathbb Z_n \to [0,\infty), \] for every $n \ge 4$. The proof is inspired by the recent work of Frank and Ivanisvili~\cite{FrankIvanisvili2026} on a sharp log-Sobolev inequality for nearest-neighbor simple random walk. We use their cubic-majorant reduction, which turns the problem into a 3rd moment estimate; the new point is a blockwise 3rd moment estimate adapted to the word-length multiplier. The same 3rd moment argument also recovers the log-Sobolev inequality for Poisson-semigroup on the circle, first proved by Weissler~\cite{Weissler1980}. The same sharp inequalities were also obtained recently by Yao~\cite{Yao2026} by a different method.

We first extend Calderón's transfer principle to weighted spaces, and then we apply our results to obtain some new weighted inequalities in ergodic theory and ergodic $H^1$ spaces.