IMPROVEMENT OF ONE INEQUALITY FOR ALGEBRAIC POLYNOMIALS

We prove that the inequality parallel to g(./n)parallel to(L1[-1,1])parallel to P(n+k)parallel to(L1[-1,1]) <= 2 parallel to gP(n+k)parallel to(L1[-1,1],) where g: [-1,1] -> R is a monotone odd function and P(n+k) is an algebraic polynomial of degree not higher than n + k, is true for all natural n for k = 0 and all natural n >= 2 for k = 1. We also propose some other new pairs (n, k) for which this inequality holds. Some conditions on the polynomial P(n+k) under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed.