A Proof of the Koumandos--Ruscheweyh Conjecture

Abstract

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.