Dual Smale's mean value conjecture for odd polynomials
Quanyu Tang
TL;DR
This work addresses the Dual Smale's mean value conjecture for complex polynomials with $P(0)=0$ and $P'(0)=1$, focusing on odd polynomials of degree $d\ge3$. The authors reduce $P$ to the form $P(z)=z\,Q(z^{2})$ with $Q(0)=1$, and study the auxiliary function $H(u)=u\,Q(u)^{2}$ alongside $R(u)=Q(u)+2uQ'(u)$, noting that critical points of $H$ lie in $\{Q=0\}$ or $\{R=0\}$ and that $H'(u)=Q(u)R(u)$. They then apply the Dubinin–Sugawa inequality, which guarantees a critical point $w$ of $H$ with $\left|\frac{H(w)}{w}\right|\ge \frac{1}{d^{2}}$, forcing $w$ into $\{R=0\}$ and yielding a bound $|Q(w)|\ge \frac{1}{d}$. From $P'(c)=0$ at $c=\pm\sqrt{w}$, they obtain $\left|\frac{P(c)}{c}\right|\ge \frac{1}{d}$, proving the conjecture for the odd polynomial class with nonzero linear term. The paper also notes that equality cannot occur in general and mentions a dual variant under symmetric scaling, reflecting connections to Ng's results. Overall, the work extends the range of polynomials for which the Dual Smale's mean value conjecture holds and introduces a constructive critical-point framework via the $H$-function.
Abstract
We prove Dual Smale's mean value conjecture for all odd polynomials with nonzero linear term. Precisely, if $P$ is an odd polynomial of degree $d\ge3$ with $P(0)=0$ and $P'(0)=1$, then there exists a critical point $ζ$ of $P$ such that $$ \left|\frac{P(ζ)}ζ\right| \ge \frac1d. $$This result can be regarded as a dual counterpart of T. W. Ng's theorem on Smale's mean value conjecture for odd polynomials with nonzero linear term [J. Aust. Math. Soc. 75 (2003), 409--411].