Markoff Type Inequalities for Curved Majorants

Abstract

Let Л_n be the set of all polynomials of degree at most n with complex coefficients. As usual denote by

$${T_m}(x)\,:\, = \cos \,(m\,arc\,cos\,x)$$

and

$${U_m}(x)\;:\; = \;{(1 = {x^2})^{ - 1/2}}\sin ((m + 1)arc\;cos\;x)$$

the Tschebyscheff polynomials of degree m of the first and second kind, respectively. Furthermore, let us write ∥·∥ for the maximum norm on the unit interval, i. e.

$$\left\| P \right\|: = \begin{array}{*{20}c} {\max } \\ { - 1 \leqq x \leqq 1} \\ \end{array} |p(x)|$$

.