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Phelps–Rodrigues extremal conjecture for Borcea's variance inequality

From papers

Let FF be a polynomial, let F′F' be its derivative, and let h(F,F′)h(F,F') and σp(F)\sigma_p(F) be as in Borcea's variance conjecture. The equality cases concern polynomials of degree at most nn.

Phelps–Rodrigues extremal conjecture. If p>1p>1, equality in Borcea's variance inequality occurs if and only if

F(z)=a(z−c)n−bF(z)=a(z-c)^n-b

for some complex numbers a,b,ca,b,c, whenever deg⁡F≤n\deg F\leq n.

The statement is an extremal refinement of Borcea's conjecture. The source gives a counterexample when p=1p=1, so the restriction p>1p>1 is essential; no resolution of the stated equality characterization is supplied here.

Progress summary

Open

No proof or counterexample has been found for the proposed equality characterization, although the corresponding claim fails when the parameter equals one.

The conjecture says that, for p>1p>1, equality occurs exactly for polynomials of the form F(z)=a(z−c)n−bF(z)=a(z-c)^n-b with deg⁡F≤n\deg F\leq n. Phelps and Rodrigues proposed the related strengthened Sendov conjecture in 1972, but the retrieved literature does not establish this precise characterization.

Known results

  • Khavinson, Pereira, Putinar, Saff, and Shimorin state the extremal conjecture and give, for p=1p=1 and n>2n>2, the counterexample F(z)=z(z−1)n−1F(z)=z(z-1)^{n-1}.
  • The same paper identifies the proposed polynomials as the only absolute maximizers of h(F,F′)h(F,F') on the normalized class, but supplies no proof for p>1p>1.
  • The inequality at p=∞p=\infty is Sendov’s conjecture, and validity at one parameter implies validity for all larger parameters.

Current status (as of August 2026): The p=1p=1 extension is disproved, while the stated p>1p>1 equality characterization remains open with no publicly retrieved proof, counterexample, or verification.

Sources
Sources & referencesView supporting material

Primary source

Dmitry Khavinson, Rajesh Pereira, Mihai Putinar, Edward B. Saff and Serguei Shimorin, “Borcea's variance conjectures on the critical points of polynomials”, arXiv:1010.5167 (2011).

complex-analysisinequalitiespolynomial-zeroes

Solutions 0

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