On some inequalities for polynomial functions

Let Pi(n) denote the set of all algebraic polynomials of degree at most n. Let P(x) = Sigma(n)(k=0) a(k)x(k) and parallel to P parallel to(d sigma) = (integral(R) vertical bar P(x)vertical bar(2)d sigma(x)(1/2), where d sigma(x) is a nonnegative measure on R. Milovanovic determined best constants C-nk such that vertical bar a(k)vertical bar <= C-nk parallel to P parallel to(d sigma), for k = 0, 1,...n. In the present work, we will propose a new way of proving the above inequality, which will lead us to finding the optimal constant C, such that parallel to P parallel to(infinity) <= C parallel to P parallel to(d sigma), where parallel to center dot parallel to(infinity) denotes the uniform norm on [0,1].