A Proof of the Koumandos--Ruscheweyh Conjecture

Abstract

We prove the Koumandos--Ruscheweyh conjecture for every $0 < ρ\leq 1$. If $μ^*(ρ)$ 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 $n\geq0$, $0 < μ\leq μ^*(ρ)$ and $z\in\mathbb{D}$, where $s_n^μ(z)=\sum_{k=0}^n(μ)_kz^k/k!$. The parameter $μ^*(ρ)$ is optimal. The proof combines an all-parameter gamma-coefficient comparison with an exact beta integral and a binomial variance estimate. These estimates reduce 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 positive-real-part conjecture follows as a sharp corollary.