MathDB

Dual Mean Value Conjecture

For every integer d≥2d\ge 2 and every polynomial P∈C[z]P\in\mathbb{C}[z] of degree dd satisfying P(0)=0P(0)=0 and P(1)=1P(1)=1, there exists a critical point ζ≠0\zeta\ne 0 with P′(ζ)=0P'(\zeta)=0 such that ∣P(ζ)ζ∣≥1d\left|\frac{P(\zeta)}{\zeta}\right|\ge \frac{1}{d}. Equivalently, max⁡P′(ζ)=0, ζ≠0∣P(ζ)ζ∣≥1d\max_{P'(\zeta)=0,\,\zeta\ne 0}\left|\frac{P(\zeta)}{\zeta}\right|\ge \frac{1}{d}.

References

Primary source

arXiv

Additional references

approximation-theorycomplex-analysis
Posted

Progress summary

Refreshed
Claimed progress

A new result strengthens the best bound in every dimension, but does not settle the conjecture.

The Dual Mean Value Conjecture seeks a universal lower bound for a normalized polynomial at one of its critical points. The full conjecture remains open.

Known results

  • The conjecture is proved through degree n=7n=7; degrees 2≤n≤42 \le n \le 4 were known earlier, degrees 55 and 66 were proved by Hinkkanen, Kayumov, and Khammatova, and degree 77 was proved in 2023.
  • For general degree nn, the cited universal bound is max⁡P′(ζ)=0∣P(ζ)/ζ∣≥1/n2\max_{P'(\zeta)=0}\left|P(\zeta)/\zeta\right|\ge 1/n^2 (Dubinin).

August 2026 stronger lower bound

A paper reported on August 27, 2026, improves Dubinin’s unconditional lower bound for all d≥2d \ge 2 using logarithmic capacities of polynomial lemniscates; the improvement is strictly stronger but does not solve the conjecture. Separately, Q. Tang’s October 19, 2025 preprint claims the conjecture for all odd polynomials with nonzero linear term, including max⁡P′(ζ)=0∣P(ζ)/ζ∣≥1/d\max_{P'(\zeta)=0}\left|P(\zeta)/\zeta\right|\ge 1/d; this claim is unverified.

Current status (as of August 2026): The conjecture is settled through degree 77 and has stronger general lower bounds, but remains open in full generality; the odd-polynomial result is unverified.

Sources

Solutions 0

No solutions have been posted yet.