Coefficient bounds for polynomials that map the unit interval to itself

Question

While studying the "coin-flipping degree" problem I have come across the following conjecture. It gives bounds on the power coefficients of a polynomial that maps the unit interval to itself. If true, this could contribute to solving the "coin-flipping degree" problem.

Conjecture: Let $p(\lambda)=a_0 \lambda^0 + ... + a_n\lambda^n$ be a polynomial that maps the closed unit interval to itself and satisfies $0\lt p(\lambda)\lt 1$ whenever $0\lt\lambda\lt 1$. Then $|a_i|\le |b_i|$, where $b_i$ is a power coefficient of the following polynomial: $$q(\lambda) = b_0 \lambda^0 + ... + b_n\lambda^n = (T_n(1-2\lambda)+1)/2,$$ and where $T_n(x)$ is the Chebyshev polynomial of the first kind of degree $n$.

Has this conjecture been studied or proved anywhere? If not, is the conjecture true?

Answer 1

This is correct. If $0=c_0<c_1<\ldots<c_n=1$ are extrema of the polynomial $q$, that is, $q(c_j)=(-1)^j$ for $j=0,1,\ldots,n$, you may interpolate $p$ at nodes $c_0,\ldots,c_n$ to get $$ p(x)=\sum_{j=0}^n p(c_j)\frac{\prod_{k\ne j}(x-c_k)}{\prod_{k\ne j}(c_j-c_k)}, $$ that gives for the coefficient $a_m$ (for $m=0,1,\ldots,n$) the formula $$ a_m=\sum_{j=0}^n p(c_j)\frac{[x^m]\prod_{k\ne j}(x-c_k)}{\prod_{k\ne j}(c_j-c_k)}=\sum_{j=0}^n p(c_j)\cdot \alpha_{m,j}, $$ where the sign of $\alpha_{m,j}(-1)^j$ is the same for all $j=0,1,\ldots,n$. Thus $|a_m|$ under constraints $|p(c_j)|\leqslant 1$ is maximal for $p=q$.