A Proof of the Koumandos--Ruscheweyh Conjecture
Abstract
We prove the Koumandos--Ruscheweyh conjecture for every 0<ρ≤1. If ν(ρ) is the unique root in (0,1] of ∫0(1+ρ)πtμ−1sin(t−ρπ) dt=0, then (1−z)ρsnμ(z)≺((1+z)/(1−z))ρ for every n≥0, 0<μ≤ν(ρ) and z∈𝔻, where snμ(z)=∑k=0n(μ)kzk/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.
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BibTeX
@misc{ma2026proof,
title = {{A Proof of the Koumandos--Ruscheweyh Conjecture}},
author = {Yicen Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02285},
eprint = {2610.02285},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02285},
}
RIS
TY - UNPB
TI - A Proof of the Koumandos--Ruscheweyh Conjecture
AU - Yicen Ma
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.02285
UR - https://arxiv.org/abs/2610.02285
AB - We prove the Koumandos--Ruscheweyh conjecture for every 0<ρ≤1. If ν(ρ) is the unique root in (0,1] of ∫_0^(1+ρ)πt^μ−1sin(t−ρπ) dt=0, then (1−z)^ρs_n^μ(z)≺((1+z)/(1−z))^ρ for every n≥0, 0<μ≤ν(ρ) and z∈𝔻, where s_n^μ(z)=∑_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.
ER -