Dual Smale's mean value conjecture for odd polynomials

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 $d\ge 3$ with $P(0)=0$ and $P^{\mathrm{\prime }}(0)=1$, then there exists a critical point $\zeta$ of $P$ such that $$\mid \frac{P(\zeta )}{\zeta }\mid \ge \frac{1}{d}\mathrm{.}$$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].