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 $P(0)=0$, $P^{\prime }(0)=1$, let $S(P)={\min }_{P^{\prime }(c)=0}|P(c)/c|$ and let $\mathcal{M}={\sup }_{P\in \mathcal{P}}S(P)$ be Smale's mean value constant. Smale proved that $\mathcal{M}\le 4$, observed that $\mathbb{M}\ge 1$, 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 $\mathrm{deg}B\le d$ 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 $S(P)>1.1018$. The polynomial construction behind these results also settles Problems 1A and 1B of Smale's 1981 paper. For $P\in \mathcal{P}$ and an integer $k\ge 2$ let $S_k(P)={\min }_{P^{\prime }(c)=0}|P^{(k)}(0)/k!{|}^{1/(k-1)}|P(c)|$. Smale proved, by Löwner's coefficient estimate for inverses of univalent functions, that ${\sup }_{P\in \mathcal{P}}{S}_k(P)\le C_k^{1/(k-1)}<4$, where $C_k$ is the $k$-th Catalan number and he asked whether the constant $4$ (Problem 1A), or the constant $2=C_2$ for $k=2$ (Problem 1B), can be reduced. We prove that ${\sup }_{P\in \mathcal{P}}{S}_k(P)=C_k^{1/(k-1)}$ for every $k\ge 2$. Hence the constants $4$ and $2$ cannot be reduced and the answers to both problems are negative.

arXiv commentThis version includes the solution of Problem 1A and 1B and Smale's 1981 paper