A Proof of the Koumandos--Ruscheweyh Conjecture

Abstract

We prove the Koumandos--Ruscheweyh conjecture for every $0<\rho \le 1$. If $\nu (\rho )$ is the unique root in $(0,1]$ of ${\int }_0^{(1+\rho )\pi }{t}^{\mu -1}\sin (t-\rho \pi )\mathrm{d}t=0$, then $(1-z{)}^{\rho }{s}_n^{\mu }(z)\prec ((1+z)/(1-z){)}^{\rho }$ for every $n\ge 0$, $0<\mu \le \nu (\rho )$ and $z\in \mathbb{D}$, where $s_n^{\mu }(z)={\sum }_{k=0}^n(\mu {)}_k{z}^k/k!$. The parameter $\nu (\rho )$ 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.