Let $rho in )0,1($, $varepsilonin(0,rho)$, $k in mathbb{N}^*$ and

$$P^varepsilon_k(X) = tfrac{T_kleft(tfrac{2(X+varepsilon)}{rho+varepsilon}-1 right)}{left|T_kleft(tfrac{2(1+varepsilon)}{rho+varepsilon}-1 right) right|}$$

where $T_k$ is the Chebyshev polynomial of first kind with degree $k$, i.e. $T_k(cos(theta)) = cos(ktheta),; forall theta in mathbb{R}$.

I would like to show that the function $varepsilon rightarrow | P^varepsilon_k |_1$ (with $|cdot|_1$ is the sum of the absolute value of the polynomial’s coefficients) is decreasing on $(0,rho)$. It is true for $varepsilon$ near $0$ and seems true numerically for all $varepsilonin(0,+infty($.

### Case $varepsilon$ close to 0

We have from the definition of $T_k$ that its roots are the $(z_i)_{iin(0 ,k-1)} = (cosleft(tfrac{2i+1}{2k}pi right))_{iin(0 ,k-1)} in (-1,1)$. The roots of $P^varepsilon_k(X)$ are the $z^varepsilon_i$ defined such that $tfrac{2z^varepsilon_i+2varepsilon}{rho+varepsilon} -1 = z_i $. This corresponds to

$$ z^varepsilon_i = tfrac{(rho+varepsilon)z_i + rho – varepsilon}{2} in (-varepsilon,rho), quad i=0,dots,k-1.$$

The smallest root $z^varepsilon_{k-1} = tfrac{(rho+varepsilon)cos(tfrac{2k-1}{2k}pi) + rho – varepsilon}{2}$ is nonnegative for $varepsilonin(0,rhotfrac{1+cos(tfrac{2k-1}{2k}pi)}{1-cos(tfrac{2k-1}{2k}pi)})$. This means that this for choice of $varepsilon$, all the roots of $P^varepsilon_k$ are nonnegative and therefore coefficients of $P^varepsilon_k$ alternate signs (we see that by expressing the coefficients using the roots). This fact implies that one can write $| P^varepsilon_k|_1 = left | P^varepsilon_k(-1) right|$ which can be expressed as

begin{align*}

left | P^varepsilon_k(-1) right| &= tfrac{left|T_kleft(tfrac{2(-1+varepsilon)}{rho+varepsilon} -1right)right|}{T_kleft(tfrac{2(1+varepsilon)}{rho+varepsilon} -1right)}\

& = tfrac{left(1 +tfrac{rho-varepsilon}{2} -sqrt{(1+rho)(1-varepsilon)}right)^k +left(1 +tfrac{rho-varepsilon}{2} +sqrt{(1+rho)(1-varepsilon)}right)^k}{left(1 +tfrac{rho-varepsilon}{2} -sqrt{(1-rho)(1+varepsilon)}right)^k +left(1 +tfrac{rho-varepsilon}{2} +sqrt{(1-rho)(1+varepsilon)}right)^k},

end{align*}

where we use the formula $T_k(x) = tfrac{1}{2}left(left(x – sqrt{x^2-1} right)^k+left(x + sqrt{x^2-1} right)^k right)$ for $|x| geq 1$. Finally, a simple study of function shows that the numerator is decreasing and the denominator increasing.

### Additional info

Note that $P^varepsilon_k$ is solution of the optimization problem

$$ underset{substack{pinmathbf{R}_k(X)\p(1)=1}}{min}underset{xin(-varepsilon,rho)}{max}|p(x)|. $$

Any ideas or pointers are welcome, thanks for your help.