Sharp constants in the Smale mean value conjecture for finite Blaschke products

For a finite Blaschke product $B$ of degree $n\ge2$ with $B(0)=0$ and $B'(0)\ne0$, let $S(B)=\min\{|B(\zeta)/(\zeta B'(0))|: B'(\zeta)=0,\ |\zeta|<1\}$ be the smallest normalized critical-value quotient over the critical points of $B$ in the unit disk, and $T(B)=\max\{|B(\zeta)/(\zeta B'(0))|: B'(\zeta)=0,\ |\zeta|<1\}$ its dual. Ng and Zhang (2016) introduced the hyperbolic analogue of Smale's mean value conjecture via the constants $K_n=\sup S(B)$ and $L_n=\inf T(B)$ and proved only bounds: $1\le K_n\le 2^{(2n-1+(2n-3)4^{1/(1-n)})/(2n-1)}$ (a constant of the form $4-O(1/n)$) and $1/4^n<L_n\le1/n$. Dubinin (2021–2024) determined the sharp bound for the un-normalized quantity $\min|B(\zeta)|$ at fixed $c=|B'(0)|$, with the Chebyshev–Blaschke extremal $B_c(z)=z(z^{n-1}-c)/(1-cz^{n-1})$, and conjectured that the same extremal governs the $\zeta B'(0)$-normalized quotient. The open problem is to determine the sharp constants exactly — the fixed-$c$ function $f_n(c)$, the universal constants $K_n$ and $L_n$ — or, in its sharpest specific form, to prove or refute Dubinin's conjecture, whose positive resolution would imply Smale's 1981 polynomial conjecture $M_n=(n-1)/n$.

Background

In his 1981 study of the efficiency of Newton's method for finding zeros of a complex polynomial, Steve Smale asked whether the following mean value property holds: for every polynomial $P$ of degree $n\ge2$ with $P(0)=0$ and $P'(0)\ne0$, there exists a critical point $\zeta$ of $P$ (a zero of $P'$) such that $|P(\zeta)/(\zeta P'(0))|\le(n-1)/n$. The constant $(n-1)/n$ is best possible, attained by $P(z)=cz+z^n$; writing $M_n$ for the supremum over all degree-$n$ polynomials of this minimum quotient, the conjecture $M_n=(n-1)/n$ has remained open since 1981. It is known to hold for $n\le4$, and the best general upper bounds are of the form $4-O(1/\sqrt n)$. The dual conjecture asks for a critical point with $|P(\zeta)/(\zeta P'(0))|\ge1/n$; it has been proved analytically through $n=7$ but is open in general.

A finite Blaschke product is a rational function $B(z)=\alpha\prod_{k=1}^{n}(z-z_k)/(1-\bar{z}_k z)$ with $|\alpha|=1$ and all $|z_k|<1$; these are exactly the $n$-to-$1$ analytic self-maps of the unit disk, and they have been viewed as the non-Euclidean (disk) counterparts of polynomials since Walsh, a viewpoint propagated by Beardon and Minda and developed into a systematic dictionary by Ng and Tsang. Motivated by this dictionary, Ng and Zhang (2016) formulated Smale's mean value conjecture and its dual for finite Blaschke products with $B(0)=0$ and $B'(0)\ne0$: defining $S(B)=\min\{|B(\zeta)/(\zeta B'(0))|: B'(\zeta)=0,\ |\zeta|<1\}$, $T(B)=\max\{|B(\zeta)/(\zeta B'(0))|: B'(\zeta)=0,\ |\zeta|<1\}$, $K_n=\sup S(B)$ and $L_n=\inf T(B)$ over degree-$n$ products (Sheil-Small had already shown $M_n\le K_n$, so the Blaschke constants dominate the polynomial ones), they proved $K_n\le 2^{(2n-1+(2n-3)4^{1/(1-n)})/(2n-1)}$ for $n\ge2$, $K_n\ge1$ for $n\ge3$ with $K_2=1$ and no extremal product, $L_n>1/4^n$ for $n\ge2$, and $L_n\le1/n$ for $n\ge3$ with $L_2=1/2$ and no extremal. In particular the literal transfer of the polynomial constant $(n-1)/n$ provably fails for Blaschke products, and no sharp value is known for any $n\ge3$. A structural obstruction is that the Blaschke class is closed under neither differentiation nor finite sums, so most polynomial techniques do not transfer.

Dubinin then developed a program of condenser-capacity and symmetrization methods for Blaschke critical values (condenser capacities are conformally invariant energy-type quantities attached to ring domains, and symmetrization replaces a configuration by a more symmetric one without decreasing the relevant capacity). He first (2021) announced sharp bounds for both the least and the greatest critical values of Blaschke products, depending only on $n$ and $|B'(0)|$ but without the $\zeta B'(0)$ normalization. He then (2022) proved the upper bound: at fixed $c=|B'(0)|\in(0,1)$ the sharp supremum of $\min\{|B(\zeta)|: B'(\zeta)=0\}$ is attained by the Chebyshev–Blaschke product $B_c(z)=z(z^{n-1}-c)/(1-cz^{n-1})$ (so named by analogy with Chebyshev polynomials, the corresponding extremal objects on the polynomial side); as a polynomial corollary he obtained $\min\{|P(\zeta)|: P'(\zeta)=0\}\le(n-1)(|P'(0)|/n)^{n/(n-1)}$, answering a question of Smale. Finally (2024) he conjectured that the same extremal governs the normalized quantity: for every degree-$n$ Blaschke product with $B(0)=0$ and $|B'(0)|=c$, some critical point $\zeta$ satisfies $|B(\zeta)/(\zeta B'(0))|\le f(c)$, where $f(c)$ is the common value of $|B_c(\zeta)/(\zeta B_c'(0))|$ at the critical points of $B_c$ in the unit disk — explicitly, these are the $n-1$ solutions of $\zeta^{n-1}=\tau$, where $\tau\in(0,c)$ is the unique root in $(0,1)$ of $\tau^2+((n-2)c-n/c)\tau+1=0$, and $f(c)=(c-\tau)/(c(1-c\tau))$; his Lemma 1 shows $f$ increases from $(n-1)/n$ (as $c\to0$) to $1$ (as $c\to1$). His Theorem 1 proves only a weaker bound carrying an explicit un-absorbed slack factor, and he observes that the conjecture would imply Smale's original polynomial conjecture $M_n=(n-1)/n$. The 2023 Uspekhi survey by Avkhadiev–Kayumov–Nasyrov confirms that neither the polynomial constants nor the Blaschke constants are determined.

Problem Statement

For each degree $n\ge2$ and each $c\in(0,1)$, determine exactly the sharp constant

$$f_n(c)=\sup\left\{\min_{B'(\zeta)=0,\ |\zeta|<1}\left|\frac{B(\zeta)}{\zeta B'(0)}\right|\ :\ B\ \text{a finite Blaschke product of degree}\ n\ \text{with}\ B(0)=0, |B'(0)|=c\right\},$$

and consequently the universal hyperbolic Smale constant $K_n=\sup_{0<c<1}f_n(c)$; dually, determine the sharp constant $L_n=\inf\{\max_{B'(\zeta)=0,\ |\zeta|<1}|B(\zeta)/(\zeta B'(0))|:\ \deg B=n,\ B(0)=0,\ B'(0)\ne0\}$. The sharpest specific form of the question is Dubinin's conjecture that $f_n(c)=f(c)$ for all $n\ge2$ and $c\in(0,1)$, where $B_c(z)=z(z^{n-1}-c)/(1-cz^{n-1})$ is the Chebyshev–Blaschke product and $f(c)=(c-\tau)/(c(1-c\tau))$, with $\tau\in(0,c)$ the unique root in $(0,1)$ of $\tau^2+((n-2)c-n/c)\tau+1=0$; equivalently, $f(c)$ is the common value of $|B_c(\zeta)/(\zeta B_c'(0))|$ at the $n-1$ critical points of $B_c$ in the unit disk (the solutions of $\zeta^{n-1}=\tau$). The conjecture would give $K_n=1$ (a supremum not attained), while the expected dual value is $L_n=1/n$. The primal sub-claim (determining $f_n(c)$, and hence $K_n=\sup_{0<c<1}f_n(c)$) and the dual sub-claim (determining $L_n$) are independent — the primal bounds the minimum over critical points whereas the dual bounds the maximum, so neither implies the other — and a complete resolution of the task is any one of the following, each settling its own sub-claim and leaving the other open: (a) a proof that $f_n(c)=f(c)$ for all $n\ge2$ and $c\in(0,1)$ (which would in particular give $K_n=1$); (b) a counterexample refuting this; (c) a proof that $L_n=1/n$; (d) a counterexample showing $L_n<1/n$. On the positive branch a complete answer must identify the exact optimal values with rigorous upper-bound arguments covering the whole class and matching extremal products or extremizing sequences, including the endpoint behavior as $c\to0$ and $c\to1$; on the refutation branch a counterexample must exhibit, for some $n\ge2$ and $c\in(0,1)$, an admissible Blaschke product $B$ with $B(0)=0$ and $|B'(0)|=c$ such that every critical point $\zeta$ of $B$ in the unit disk satisfies $|B(\zeta)/(\zeta B'(0))|>f(c)$ — or, dually, a degree-$n$ product with $\max_{B'(\zeta)=0,\ |\zeta|<1}|B(\zeta)/(\zeta B'(0))|<1/n$, showing $L_n<1/n$.

References

1. Tuen-Wai Ng, Yongquan Zhang, 'Smale's mean value conjecture for finite Blaschke products', The Journal of Analysis 24 (2016) 331–345. DOI: 10.1007/s41478-016-0007-4, https://doi.org/10.1007/s41478-016-0007-4 (preprint arXiv:1609.00170, https://arxiv.org/abs/1609.00170)
2. Steve Smale, 'The fundamental theorem of algebra and complexity theory', Bulletin of the American Mathematical Society 4 (1981) 1–36. DOI: 10.1090/S0273-0979-1981-14858-8, https://doi.org/10.1090/S0273-0979-1981-14858-8
3. V. N. Dubinin, 'Critical Values of Finite Blaschke Products', Doklady Mathematics 104 (2021) 163–164. DOI: 10.1134/S1064562421040050, https://doi.org/10.1134/S1064562421040050
4. V. N. Dubinin, 'An upper bound for the least critical values of finite Blaschke products', Sbornik: Mathematics 213 (2022) 744–751. DOI: 10.1070/SM9679, https://doi.org/10.1070/SM9679 (Russian original: Matematicheskiĭ Sbornik 213:6 (2022) 13–20)
5. V. N. Dubinin, 'Blaschke Products and Smale's Conjecture on Complex Polynomials', Mathematical Notes 116 (2024) 252–257. DOI: 10.1134/S0001434624070198, https://doi.org/10.1134/S0001434624070198 (Russian original: Matematicheskie Zametki 116:2 (2024) 229–235, DOI: 10.4213/mzm14315, https://doi.org/10.4213/mzm14315)
6. F. G. Avkhadiev, I. R. Kayumov, S. R. Nasyrov, 'Extremal problems in geometric function theory', Russian Mathematical Surveys 78:2 (2023) 211–271. DOI: 10.4213/rm10076e, https://doi.org/10.4213/rm10076e (Russian original: Uspekhi Matematicheskikh Nauk 78:2 (2023) 3–70, DOI: 10.4213/rm10076, https://doi.org/10.4213/rm10076)
7. Aimo Hinkkanen, Ilgiz R. Kayumov, Diana M. Khammatova, 'Dual Smale's Mean Value Conjecture for n=7', Constructive Approximation 61 (2025) 603–613. DOI: 10.1007/s00365-025-09709-4, https://doi.org/10.1007/s00365-025-09709-4