Inequalities involving generalized interpolation polynomials

The generalized Vandermonde determinant and the generalized interpolation polynomial are defined. We establish inequalities involving generalized interpolation polynomials under the proper hypotheses. Our main result is: |∑j=1nVn(x(j)(u),α)Vn(x,α)yj|≤[2n(n−1)∑i=0[n−12]∑j=1[n−12]1i+j]n(n−1)2⋅vn(xnθ−x1θθxnθα)∑j=1n|yj|Vn(xxn,α). The approach used is able to combine algebraic and analytical theories.