Interpolation with lagrange polynomials a simple proof of Markov inequality and some of its generalizations

The following theorem is provedTheorem 1. Let q be a polynomial of degree n (q∈P_n) with n distinct zeroes lying in the interval ?1,1] and $$|p^{(k)} (x)| \leqslant \max \{ |q^{(k)} (x)|,|\frac{1}{k}(x^2 - 1)q^{(k + 1)} (x) + xq^{(k)} (x)|\} .$$ If polynomial p∈P_n satisfies the inequality $$|p(\tau _l )| \leqslant |q(\tau _l )|,{\text{ }}\tau _l \in \Delta '_{q'} $$ then for each \(k = \overline {1,n} \) and any x∈?1,1] its k-th derivative satisfies the inequality $$||p_n^{(k)} || \leqslant ||T_n^{(k)} ||$$ This estimate leads to the Markov inequality for the higher order derivatives of polynomials if we set q=T_n, where T_n is Chebyshev polynomial least deviated from zero. Some other results are established which gives evidence to the conjecture that under the conditions of Theorem 1 the inequality ‖p^(k)‖≤‖q^(k)‖ holds.