Chebyshev expansion positivity for snake polynomials under even convex majorants

Establish that for every continuous, even, convex majorant μ ≥ 0 on [−1, 1], the snake polynomial ω_μ associated with μ has a non-negative expansion in the Chebyshev polynomials of the first kind.

Background

Snake polynomials are extremal polynomials oscillating maximally between ±μ under the constraint |P(x)| ≤ μ(x). Their alternation points and Chebyshev expansions relate to extremal inequalities for polynomial derivatives (Markov and Duffin–Schaeffer inequalities).

If a snake polynomial admits a non-negative or sign-alternating Chebyshev expansion, it is extremal for both Markov- and Duffin–Schaeffer-type inequalities under the majorant μ. The conjecture proposes that this positivity should hold whenever μ is continuous, even, and convex.