AN INEQUALITY OF DUFFIN-SCHAEFFER-SCHUR TYPE

Abstract

It is shown here that the transformed Chebyshev polynomial of the second kind ${\bar{U}}_n(x):=U_n(x\cos \frac{\pi }{n+1})$ 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 $\{\cos \frac{j\pi }{n}/\cos \frac{\pi }{n+1}{\}}_{j=1}^{n-1}$.