The famous Chebyshev polynomials satisfy many extremal properties. One of these is that they attain the largest possible derivative over the interval [-1,1] among polynomials whose absolute value over that interval is at most 1. (For degree $k$ polynomials, the maximum derivative is $k^2$.)
Let the $\ell_1$ of a polynomial be the sum of absolute values of its coefficients. We know that the leading coefficient of the degree $k$ Chebyshev polynomials is $2^{k-1}$. Thus, the $\ell_1$ of the degree $k$ Chebyshev polynomial is at least exponential in the degree.
Does anyone know whether this is necessary? Must degree $k$ polynomials that have absolute value at most 1 in the interval [-1,1] and attain a derivative of $\Omega (k^2)$ in the interval have $\ell_1$ that is exponential in the degree?
It is easy to prove that the $\ell_1$ must be at least $k$, but I suspect one can do much better.