Sharp constants in Crane's Möbius-invariant mean value conjectures for rational maps

Smale's mean value conjecture (1981) for complex polynomials is affine-invariant and therefore does not extend verbatim to rational maps of the Riemann sphere, which treat the point at infinity specially. Crane (2006) proposed two Möbius-invariant generalizations to rational maps: a two-point version (given two points x, y, find a critical point zeta and a Möbius map M such that M∘R fixes x, y, zeta and the multiplier of M∘R at x is large) and a one-point version using two critical points. He proved both only with the constants 1/4 and 1/2 — the rational-map analogues of Smale's factor 4 — and conjectured that in both the largest admissible constant, as a function of the degree n, is n/(n−1), the value forced by the Newton map of (z−c)^n+b(z−c), which is Möbius-equivalent to the extremal polynomial (z^n−nz)/(1−n). No subsequent work addresses the sharp constants or the extremal rational maps. The problem is open in its natural generality; degree 2 is settled and degree 3 is the first open case.

Background

Let $\hat{\mathbb{C}}$ be the Riemann sphere and let $R:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ be a rational map of degree $n\ge 2$ (a quotient of two polynomials with no common factor, of maximal degree $n$). A critical point of $R$ is a point at which the local degree of $R$ exceeds one — equivalently a zero of the derivative in any local coordinate — and its image is a critical value. A rational map of degree $n$ has $2n-2$ critical points counted with multiplicity. If $x$ is a fixed point of a holomorphic self-map of $\hat{\mathbb{C}}$, its multiplier is the derivative of the map at $x$ computed in any local coordinate; the value is independent of the coordinate, so the multiplier is a conformal invariant. Möbius maps are the biholomorphic self-maps of $\hat{\mathbb{C}}$, i.e. the fractional-linear maps $z\mapsto(az+b)/(cz+d)$.

In his 1981 study of the complexity of polynomial root-finding, Smale proved that for every polynomial $p$ of degree $n\ge 2$ and every non-critical point $x$ there is a critical point $\zeta$ with

$$\left|\frac{p(\zeta)-p(x)}{(\zeta-x)\,p'(x)}\right|\le 4,$$

a complex-variable relative of the mean value theorem, and he conjectured that the factor $4$ can be improved to $1-1/n$, which is best possible for $p_0(z)=(z^n-nz)/(1-n)$ at $x=0$. The conjecture is known only for degrees $n\le 5$; the best general upper bound is $K(d)<4-2.263/\sqrt{d}$ for $d\ge 8$ (Crane), so a factor-$4$ gap remains even in the polynomial case.

The quantity $S(p,x)=\min_{p'(\zeta)=0}|(p(\zeta)-p(x))/((\zeta-x)p'(x))|$ is invariant under the two-sided affine action $p\mapsto A\circ p\circ B$ because it singles out the point $\infty$: the affine group is exactly the stabilizer of $\infty$ in the Möbius group. Crane therefore asked how to state a mean value property for rational maps that is invariant under the full Möbius group on both sides. His answer is to trade the affine normalization for a Möbius normalization: instead of comparing $R(x)$ and $R(\zeta)$ through affine quantities, one post-composes $R$ with a Möbius map $M$ chosen so that $M\circ R$ fixes the finite configuration of points under consideration, and one estimates the multiplier of $M\circ R$ at the distinguished point. This produces two conjectures.

The first (Crane's Theorem 1, proved with constant $1/4$ by an argument using the Koebe one-quarter theorem — the classical statement that a univalent, i.e. injective holomorphic, function on the unit disc maps it onto a region containing the disc of radius one quarter about the image of the centre, the same tool behind Smale's factor $4$) involves two given points: if $R(x)\ne R(y)$ and $x$ is not critical, then some critical point $\zeta$ and some Möbius map $M$ with $M\circ R$ fixing $x$, $y$ and $\zeta$ satisfy $|(M\circ R)^{\#}(x)|>1/4$. In the polynomial case with $y=\infty$ this is a reformulation of Smale's theorem. The second (Crane's Theorem 2, proved with constant $1/2$ via the logarithmic capacity of continua of given diameter — logarithmic capacity is a conformally invariant measure of the size of a compact plane set, and a continuum, i.e. a compact connected set, of diameter $d$ has capacity at most $d/2$ — together with the Koebe theorem) involves one given point and two critical points: if $R(x)$ is not a critical value, then some pair of critical points $\zeta,\kappa$ is such that, for the Möbius map $M$ making $x$, $\zeta$, $\kappa$ fixed points of $M\circ R$, one has $|(M\circ R)^{\#}(x)|\ge 1/2$; unlike Theorem 1 this does not contain Smale's polynomial theorem as a special case. Crane asks for the largest constants $K_1$ and $K_2$ (as functions of $n=\deg R$) that can replace $1/4$ and $1/2$; the example $p_0$ with $x=0$, $y=\infty$ forces $K_1,K_2\le n/(n-1)$, and he conjectures equality in both.

The candidate extremal maps are rational maps all of whose critical points are fixed (critically fixed maps). For these the normalizing Möbius map in either theorem must be the identity whenever the fixed configuration consists of fixed points, so the multiplier at any remaining fixed point directly bounds $K_1$ and $K_2$ from above. The basic examples are Newton maps $R_h(z)=z-h(z)/h'(z)$ of polynomials $h$ whose second-derivative roots are also roots of $h$, in particular $h(z)=a(z-c)^n+b(z-c)$, whose Newton map is Möbius-conjugate to $p_0$ and has multiplier exactly $n/(n-1)$ at the extra fixed point. Crane proves that non-critical fixed points of critically fixed maps are repelling (their multipliers have modulus greater than one), characterizes Newton–Raphson maps of rational functions by an integrality condition on $1/(1-R^{\#}(\chi))$ at fixed points, shows that Newton maps of rational functions cannot push the multiplier below $n/(n-1)$, and proves a forbidden-disc restriction (his Theorem 3): for a critically fixed map of degree $n$ with $m$ distinct critical points, the multiplier at any non-critical fixed point avoids the closed disc with diameter $[1,\,1+2/(n+m-2)]$, except for the extremal case $m=n$ where the remaining fixed point has multiplier exactly $n/(n-1)$. He also notes that, unlike the polynomial case (where the extremal is conjectured unique up to affine equivalence and this is known for $n\le 5$), there are probably several extremal maps per degree up to Möbius equivalence, and that for degree $2$ the problem reduces via the two-sided Möbius invariance to $R(z)=z^2$ with $x=1$, where the multiplier equals $2=n/(n-1)$, so degree $2$ is settled; the first open degree is $3$, whose parameter space is three-dimensional over $\mathbb{C}$.

Problem Statement

Let $\hat{\mathbb{C}}$ denote the Riemann sphere and let $R:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ be a rational map of degree $n\ge 2$ (a quotient of two coprime polynomials, of maximal degree $n$). As in the source, write $f^{\#}(x)$ for the multiplier of a self-map $f$ at a fixed point $x$: the derivative of $f$ at $x$ in any local coordinate, a value independent of the coordinate. A critical point of $R$ is a point at which the local degree of $R$ exceeds one, and its image is a critical value; a rational map of degree $n$ has $2n-2$ critical points counted with multiplicity. Möbius maps are the fractional-linear biholomorphic self-maps $z\mapsto(az+b)/(cz+d)$ of $\hat{\mathbb{C}}$, and two rational maps are Möbius equivalent if one is obtained from the other by pre- and post-composition with Möbius maps. Determine the sharp constants in Crane's two Möbius-invariant mean value inequalities for rational maps, as follows.

Part 1 (two-point constant $K_1$). For points $x,y\in\hat{\mathbb{C}}$ with $R(x)\ne R(y)$ and $x$ not a critical point of $R$, define

$$m_1(R,x,y)=\sup\Big\{\big|(M\circ R)^{\#}(x)\big| : \zeta \text{ ranges over critical points of } R \text{ with } \zeta\notin\{x,y\},\ M \text{ over Möbius maps with } M\circ R \text{ fixing } x,\ y,\ \zeta\Big\},$$

where, for a critical point $\zeta\notin\{x,y\}$ (the exclusion of $\zeta=x$ is automatic, since $x$ is not critical), a Möbius map $M$ with $M\circ R$ fixing $x$, $y$ and $\zeta$ exists precisely when $R(\zeta)\notin\{R(x),R(y)\}$, and is then the unique one satisfying $M(R(x))=x$, $M(R(y))=y$, $M(R(\zeta))=\zeta$; a critical point with $R(\zeta)\in\{R(x),R(y)\}$ admits no such $M$ (Möbius maps are injective) and contributes nothing to the supremum. The restriction $\zeta\ne y$ follows Crane's formulation: the proof of his Theorem 1 produces a critical point with $R(\zeta)\notin\{R(x),R(y)\}$, and if $\zeta=y$ were admitted the normalization would degenerate to two fixed points and the extremal example below would no longer force any bound. Let $K_1(n)=\inf m_1(R,x,y)$ over all rational maps $R$ of degree $n$ and all admissible pairs $(x,y)$. Crane's Theorem 1 gives $m_1(R,x,y)>1/4$ for every admissible triple, so $K_1(n)\ge 1/4$; and the map $p_0(z)=(z^n-nz)/(1-n)$ with $x=0$, $y=\infty$ (where every critical point other than $y=\infty$ itself is fixed, so $M$ must be the identity) shows $K_1(n)\le n/(n-1)$. Determine $K_1(n)$ exactly; the conjecture is $K_1(n)=n/(n-1)$ for every $n\ge 2$.

Part 2 (one-point constant $K_2$). For $x\in\hat{\mathbb{C}}$ with $R(x)$ not a critical value of $R$, define

$$m_2(R,x)=\sup\Big\{\big|(M\circ R)^{\#}(x)\big| : \zeta,\kappa \text{ range over pairs of distinct critical points of } R,\ M \text{ is the Möbius map with } M\circ R \text{ fixing } x,\ \zeta,\ \kappa\Big\},$$

where, since $R(x)$ is not a critical value, $\zeta,\kappa\ne x$ automatically; a pair with $R(\zeta)=R(\kappa)$ admits no such $M$ and contributes nothing, and when $R(\zeta)\ne R(\kappa)$ the map $M$ is the unique Möbius map with $M(R(x))=x$, $M(R(\zeta))=\zeta$, $M(R(\kappa))=\kappa$. Distinctness of the pair mirrors Crane's Theorem 2, whose proof selects $\zeta,\kappa$ with $R(\zeta)\ne R(\kappa)$; admitting $\zeta=\kappa$ would likewise degenerate the normalization to two fixed points. Let $K_2(n)=\inf m_2(R,x)$ over all rational maps $R$ of degree $n$ and all points $x$ with $R(x)$ not a critical value. Crane's Theorem 2 gives $m_2(R,x)\ge 1/2$ for every admissible pair, so $K_2(n)\ge 1/2$; the same example $p_0$ with $x=0$ (all its critical points are fixed, forcing $M$ to be the identity) shows $K_2(n)\le n/(n-1)$. Determine $K_2(n)$ exactly; the conjecture is again $K_2(n)=n/(n-1)$.

The two parts form one objective — the exact sharp constants of the Möbius-invariant rational-map mean value property, in the two natural formulations posed by the source. The problem is answered by proving $K_1(n)=K_2(n)=n/(n-1)$, or by refuting either conjecture through an admissible $(R,x,y)$ or $(R,x)$ for which the corresponding supremum is strictly below $n/(n-1)$. The source also discusses the candidate extremal maps — the critically fixed rational maps, in particular the Newton maps $R_h(z)=z-h(z)/h'(z)$ of $h(z)=a(z-c)^n+b(z-c)$, which are Möbius-conjugate to $p_0$ and realize the multiplier $n/(n-1)$ — together with constraints on any extremal or counterexample (non-critical fixed points of critically fixed maps are repelling; the multiplier at a non-critical fixed point of a critically fixed degree-$n$ map with $m$ distinct critical points avoids the closed disc with diameter $[1,\,1+2/(n+m-2)]$, with the extremal exception $m=n$, multiplier exactly $n/(n-1)$; Newton maps of rational functions cannot push the multiplier below $n/(n-1)$), and expects several extremal maps per degree up to Möbius equivalence. Degree 2 is already settled (reduction to $R(z)=z^2$, $x=1$, multiplier $2=n/(n-1)$), so the first open case is $n=3$; no degree $n\ge 3$ is known.

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