Smale's mean value constant and admissible univalent Blaschke pairs

Abstract: Let $\mathcal{P}$ be the class of polynomials $P$ of degree at least $2$ with $P(0)=0$, $P'(0)=1$, let $S(P)=\min_{P'(c)=0}|P(c)/c|$ and let $\mathcal{M}=\sup_{P\in\mathcal{P}}S(P)$ be Smale's mean value constant. Smale proved that $\mathcal{M}\le4$, observed that $\mathcal{M}\ge 1$, and asked whether $\mathcal{M}=1$. We show that $\mathcal{M}$ coincides with $\mathcal{B}=\sup\frac1{B(0)},$ the supremum running over pairs $(W,B)$ in which $W$ is a normalised univalent map of the standard unit disk $\mathbb{D}$ and $B$ is a finite Blaschke product with $B(0)>0$ such that $WB$ is locally univalent in $\mathbb{D}$. Smale's question is therefore a problem in univalent function theory. We also introduce the degree-restricted constants $\mathcal{B}_d$ for $B$ with $\mathrm{deg} B \le d$ and show that $$1.1018586788\ldots=\sqrt{\frac{8081}{6656}} \le \mathcal{B}_1 < \sqrt{\frac{7}{5}}=1.1832159566\ldots$$ This implies that $\mathcal{M}>1.1018$ and we also show that there are polynomials of arbitrarily large degree with $S(P)>1.1018$.