The dual Smale mean value conjecture for complex polynomials

The dual Smale mean value conjecture — posed independently by V. N. Dubinin and T. Sugawa (2009) and by T. W. Ng — asserts that every complex polynomial $P$ of degree $n\ge2$ normalized by $P(0)=0$ and $P'(0)=1$ has a critical point $\zeta$ (a zero of $P'$) with $|P(\zeta)/\zeta|\ge1/n$; equivalently, for every polynomial of degree $n$ and every noncritical point $z$, some critical point $\zeta$ satisfies $|P(z)-P(\zeta)|/|z-\zeta|\ge|P'(z)|/n$. This is the lower-bound counterpart of Smale's 1981 mean value conjecture, and its constant $1/n$ is exactly attained by $P_a(z)=(1-(1-az)^n)/(na)$, so it is sharp. The dual conjecture is verified for degrees $n\le7$, for conservative polynomials, for polynomials whose critical points lie on a single ray, and for rotationally symmetric (in particular odd) polynomials, while the best universal lower bound for general polynomials is Dubinin's $1/n^2$ — a factor $n$ short of the conjectured bound. The open problem is the conjecture for arbitrary complex polynomials, open already for degree $8$.

Background

A critical point of a complex polynomial $P$ is a point $\zeta$ with $P'(\zeta)=0$, and $P(\zeta)$ is the corresponding critical value. In his 1981 paper on the complexity of Newton-type root-finding methods, S. Smale proved that for every polynomial $P$ of degree $n\ge2$ and every noncritical point $z$ (a point with $P'(z)\ne0$) there is a critical point $\zeta$ with

$\left|\frac{P(z)-P(\zeta)}{z-\zeta}\right|\le4\,|P'(z)|,$

and conjectured that the factor 4 can be replaced by 1 (the optimal constant is at least $1-1/n$). Because the quotient is invariant under affine changes of variable, the conjecture is equivalent, as observed by Beardon, Minda, and Ng (2002), to its normalized form: if $P(0)=0$ and $P'(0)=1$, then some critical point satisfies $|P(\zeta)/\zeta|\le1-\frac{1}{n}$. This classical upper-bound conjecture remains open in general; for odd polynomials with nonzero linear term, Ng (2003) verified it with the constant 2 in place of 1.

Around 2009, V. N. Dubinin and T. Sugawa, and independently T. W. Ng, posed the dual, lower-bound counterpart, which became known as the dual Smale mean value conjecture: if $P$ is a complex polynomial of degree $n\ge2$ with $P(0)=0$ and $P'(0)=1$, then some critical point $\zeta$ of $P$ satisfies $|P(\zeta)/\zeta|\ge\frac{1}{n}$. In pointwise form this says that for every polynomial $P$ of degree $n$ and every noncritical point $z$, the maximum over critical points of the secant-slope modulus $|P(z)-P(\zeta)|/|z-\zeta|$ is at least $\frac{1}{n}|P'(z)|$; the two forms are equivalent, since applying the normalized conjecture to the affinely normalized polynomial $\tilde P(w)=(P(z+w)-P(z))/P'(z)$, which satisfies $\tilde P(0)=0$ and $\tilde P'(0)=1$ and whose critical points are exactly $w=\zeta-z$, yields the pointwise inequality at $z$. The conjectured constant cannot be improved: for $P_a(z)=\frac{1-(1-az)^n}{na}$, which is normalized of degree $n$, the unique critical point $\zeta=1/a$ (of multiplicity $n-1$) gives $|P_a(\zeta)/\zeta|=\frac{1}{n}$ exactly; in degree 6, Hinkkanen, Kayumov, and Khammatova (2019) showed equality can occur only for this family, and the same authors (2025) proved the analogous characterization in degree 7.

The dual conjecture is known for degrees $n\le7$: the cases $2\le n\le4$ are due to Tischler (1989) and Dubinin–Sugawa (2009), the cases $n=5$ and $n=6$ to Hinkkanen–Kayumov–Khammatova (2019), and $n=7$ to the same authors (2025) — each by selecting the critical point of minimal modulus and an increasingly heavy case analysis. It also holds for conservative polynomials (Tischler: polynomials in which $P(\zeta)/\zeta$ takes the same constant value at every critical point), for polynomials whose critical points all lie on one ray (Kayumov–Khammatova 2026, via a stronger conjectured inequality that implies the dual conjecture), and for rotationally symmetric polynomials satisfying $P(\lambda z)\equiv\lambda P(z)$ for a root of unity $\lambda$ — in particular odd polynomials, those with $P(-z)\equiv-P(z)$, settled by Tang (2025). For general polynomials, Dubinin (2012, 2019) proved the pointwise lower bound with factor $\frac{1}{n}\tan\frac{\pi}{4n}$ and then, by a more complicated technique, improved the factor $\tan\frac{\pi}{4n}$ to $\frac{1}{n}$, giving the best known universal lower bound $\left|\frac{P(z)-P(\zeta)}{z-\zeta}\right|\ge\frac{1}{n^{2}}|P'(z)|$ in pointwise form, i.e. $|P(\zeta)/\zeta|\ge\frac{1}{n^{2}}$ in normalized form — a full factor of $n$ short of the conjecture.

Problem Statement

Prove or disprove the dual Smale mean value conjecture: for every complex polynomial $P$ of degree $n\ge2$ with $P(0)=0$ and $P'(0)=1$,

$\max_{P'(\zeta)=0}\left|\frac{P(\zeta)}{\zeta}\right|\ge\frac{1}{n},$

that is, there exists a critical point $\zeta$ of $P$ (a point with $P'(\zeta)=0$) such that $\left|\frac{P(\zeta)}{\zeta}\right|\ge\frac{1}{n}$. Since $P'(0)=1\ne0$, no critical point lies at the origin, so the quotient $P(\zeta)/\zeta$ is well-defined.

Equivalently, in unnormalized pointwise form: for every complex polynomial $P$ of degree $n\ge2$ and every noncritical point $z$ (a point with $P'(z)\ne0$), there is a critical point $\zeta$ with

$\left|\frac{P(z)-P(\zeta)}{z-\zeta}\right|\ge\frac{1}{n}\,|P'(z)|.$

The degrees $n\le7$, conservative polynomials (those for which $P(\zeta)/\zeta$ takes the same value at every critical point), polynomials whose critical points all lie on a single ray, and rotationally symmetric polynomials (those with $P(\lambda z)\equiv\lambda P(z)$ for a root of unity $\lambda$, in particular odd polynomials with $P(-z)\equiv-P(z)$) are already settled, so the open content is the class of arbitrary complex polynomials — the conjecture is open already for degree $n=8$. The constant $1/n$ is exactly attained by $P_a(z)=\frac{1-(1-az)^n}{na}$, so the stated constant cannot be weakened.

References

1. S. Smale, The fundamental theorem of algebra and complexity theory, Bulletin of the American Mathematical Society (New Series) 4 (1981), 1–36. DOI: 10.1090/S0273-0979-1981-14858-8, https://doi.org/10.1090/S0273-0979-1981-14858-8
2. A. F. Beardon, D. Minda, T. W. Ng, Smale's Mean Value conjecture and the hyperbolic metric, Mathematische Annalen 322 (2002), 623–632. DOI: 10.1007/s002080000184, https://doi.org/10.1007/s002080000184
3. D. Tischler, Critical points and values of complex polynomials, Journal of Complexity 5 (1989), 438–456. DOI: 10.1016/0885-064X(89)90019-8, https://doi.org/10.1016/0885-064X(89)90019-8
4. V. N. Dubinin, T. Sugawa, Dual mean value problem for complex polynomials, Proceedings of the Japan Academy, Series A, Mathematical Sciences 85 (2009), 135–137. DOI: 10.3792/pjaa.85.135, https://doi.org/10.3792/pjaa.85.135
5. V. N. Dubinin, Methods of geometric function theory in classical and modern problems for polynomials, Russian Mathematical Surveys 67 (2012), 599–684. DOI: 10.1070/RM2012v067n04ABEH004803, https://doi.org/10.1070/RM2012v067n04ABEH004803
6. V. N. Dubinin, On the Dual Mean-Value Conjecture for Complex Polynomials, Mathematical Notes 106 (2019), 133–135. DOI: 10.1134/S0001434619070149, https://doi.org/10.1134/S0001434619070149
7. T. W. Ng, Smale's mean value conjecture for odd polynomials, Journal of the Australian Mathematical Society 75 (2003), 409–412. DOI: 10.1017/S1446788700008181, https://doi.org/10.1017/S1446788700008181
8. T. W. Ng, Y. 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
9. A. Hinkkanen, I. Kayumov, D. Khammatova, Dual Smale's mean value conjecture, Proceedings of the American Mathematical Society 147 (2019), 5227–5237. DOI: 10.1090/proc/14639, https://doi.org/10.1090/proc/14639
10. A. Hinkkanen, I. R. Kayumov, D. 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
11. I. R. Kayumov, D. M. Khammatova, On an inequality related to Dual Smale's mean value conjecture, Journal of Mathematical Analysis and Applications 556 (2026), Article 130108. DOI: 10.1016/j.jmaa.2025.130108, https://doi.org/10.1016/j.jmaa.2025.130108
12. D. M. Khammatova, On an Inequality for Critical Values of $k$-Symmetric Polynomials, Mathematical Notes 120 (2026), 366–371. DOI: 10.1134/S0001434626603412, https://doi.org/10.1134/S0001434626603412
13. Q. Tang, Dual Smale's mean value conjecture for odd polynomials, arXiv:2510.16875 [math.CV] (2025). DOI: 10.48550/arXiv.2510.16875, https://arxiv.org/abs/2510.16875