Smale's mean value constant and admissible univalent Blaschke pairs

Let $\mathcal{P}$ be the class of polynomials $P$ of degree at least $2$ with $P(0)=0$, $P^{\mathrm{\prime }}(0)=1$, let $S(P)={\min }_{P^{\mathrm{\prime }}(c)=0}\mathrm{\mid }P(c)\mathrm{/}c\mathrm{\mid }$ and let $\mathcal{M}={\sup }_{P\in \mathcal{P}}S(P)$ be Smale’s mean value constant. Smale proved that $\mathcal{M}\le 4$, observed that $\mathcal{M}\ge 1$, and asked whether $\mathcal{M}=1$. We show that $\mathcal{M}$ coincides with $$\mathcal{B}=\sup \frac{1}{B(0)},$$ 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 $\mathrm{d}\mathrm{e}\mathrm{g}B\le d$ and show that $$1.1018586788\dots =\sqrt{\frac{8081}{6656}}\le {\mathcal{B}}_1<\sqrt{\frac{7}{5}}=1.1832159566\dots$$ This implies that $\mathcal{M}>1.1018$ and we also show that there are polynomials of arbitrarily large degree with $S(P)>1.1018$.