An inequality for continuous linear functionals

Let n>1n>1 be an integer, f∈Cn[a,b]f∈Cn[a,b], and A:C[a,b]→RA:C[a,b]→R a continuous linear functional which annihilates all polynomials of degree at most n−1n−1. We give sharp inequalities of the form |A(f)|≤Mk‖f(k)‖2|A(f)|≤Mk‖f(k)‖2, k=2,…,nk=2,…,n.