The Markoff-Duffin-Schaeffer inequalities abstracted

Thek th Markoff-Duffin-Schaeffer inequality provides a bound for the maximum, over the interval -1 ≤x ≤ 1, of thek th derivative of a normalized polynomial of degreen . The bound is the corresponding maximum of the Chebyshev polynomial of degreen ,T = cos(n cos-1 x ). The requisite normalization is over the values of the polynomial at then + 1 points whereT achieves its extremal values. The inequality is an equality only if the polynomial equalsT or -T . The proof uses complex variable theory. This paper deals with a well-known generalization of polynomials—namely, functions satisfying some of the oscillation and approximation properties of ordinary polynomials. In particular, the generalized Chebyshev polynomial exhibits the extremal oscillations characteristic of the classical Chebyshev polynomial. It is shown that the direct analogs of the Markoff-Duffin-Schaeffer inequalities hold in this abstract setting and that they are included as a special case. Moreover, the proof is more elementary.