AN INEQUALITY OF DUFFIN-SCHAEFFER-SCHUR TYPE

Abstract

It is shown here that the transformed Chebyshev polynomial of the second kind $\overline{U}_{n}(x) := U_{n}\big (x \cos \frac{\pi}{n+1} \big )$ has the greatest uniform norm in [-1, 1] of its $k$-th derivative ($k = 1,...,n$) among all algebraic polynomials of degree not exceeding $n$, which vanish at $\pm 1$ and whose absolute value is less than or equal to 1 at the points $\bigg\{ \cos \frac{j\pi}{n} \big/ \cos \frac{\pi}{n+1}\bigg\}_{j=1}^{n-1}$.