Koumandos--Ruscheweyh猜想的证明

中文摘要 AI 辅助

我们证明了对于所有$0<\rho\leq 1$的Koumandos--Ruscheweyh猜想。若$\mu^*(\rho)$是方程$\int_0^{(1+\rho)\pi}t^{\mu-1}\sin(t-\rho\pi)\mathrm{d}t=0$在$(0,1]$中的唯一根，则对于每个$n\geq0$、$0<\mu\leq \mu^*(\rho)$和$z\in\mathbb{D}$，有$(1-z)^\rho s_n^\mu(z)\prec((1+z)/(1-z))^\rho$，其中$s_n^\mu(z)=\sum_{k=0}^n(\mu)_kz^k/k!$。参数$\mu^*(\rho)$是最优的。证明结合了全参数gamma系数比较、精确的beta积分和二项方差估计。这些估计将无穷多个自由度约简为有限多个连续区间不等式。剩余的不等式通过256位区间算术验证；提供了有理覆盖、源代码和替代公式验证器。正实部猜想作为尖锐推论随之得出。

英文摘要

We prove the Koumandos--Ruscheweyh conjecture for every $0<ρ\leq1$. If $ν(ρ)$ is the unique root in $(0,1]$ of $\int_0^{(1+ρ)π}t^{μ-1}\sin(t-ρπ)\,\mathrm{d}t=0$, then $(1-z)^ρs_n^μ(z)\prec((1+z)/(1-z))^ρ$ for every $n\geq0$, $0<μ\leqν(ρ)$ and $z\in\mathbb{D}$, where $s_n^μ(z)=\sum_{k=0}^n(μ)_kz^k/k!$. The parameter $ν(ρ)$ is optimal. The proof combines an all-parameter gamma-coefficient comparison with an exact beta integral and a binomial variance estimate, reducing the infinitely many degrees to finitely many continuous interval inequalities. The remaining inequalities are certified by 256-bit ball arithmetic; rational covers, source code and alternative-formula verifiers are provided. The weak positive-real-part conjecture follows as a sharp corollary. We also derive sharp consequences for starlike functions and Gegenbauer polynomial sections: a full-parameter convolution subordination, the optimal starlike order for uniform partial-sum sectors, and the optimal Gegenbauer parameter and sector angle. The necessary bounds and angular sharpness are obtained from explicit kernels and interior scaling limits.

发表机构

• College of Chemistry, Zhejiang University（浙江大学化学学院）

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