While studying the "[coin-flipping degree](https://mathoverflow.net/questions/448538/bounds-on-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?



Coefficient bounds for polynomials that map the unit interval to itself
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 $p=q$.


