In this search we find a degree of best one-sided approximation of the function f that lies in weighted space (L_(p,α)-space) by interpolation-operators which based on Hermite-Fejer interpolation polynomial constructed on the zeroes of Stieltjes polynomial E_(n+1)^((λ)) and the product E_(n+1)^((λ)) P_n^((λ)) for 0≤λ≤1 and 0≤λ≤1/2 (resp.). Here we denoted of these interpolation-operators by H_(n+1)^(*,±) (f,x_n) and H_(2n+1)^(**,±) (f,x_n) where x_n is the set of zeroes of E_(n+1)^((λ)) and E_(n+1)^((λ)) P_n^((λ)) (resp.) such that P_n^((λ)) is ultra-spherical polynomial with respect to ω_λ (x)=(1-x^2 )^(λ-1⁄2).The result which we end in it that the limit of differences between H_(n+1)^(*,+) and H_(n+1)^(*,_) is zero and H_(2n+1)^(**,+),H_(2n+1)^(**,_)(resp.)i.e. (⏟((lim)()┬(n→∞) H_(n+1)^(*,+)-H_(n+1)^(*,-))=0)(resp. for H_(2n+1)^(**,±) (f,x_n)).Also in this search we shall prove inverse theorem by using equivalent result between E_n (f) and E ̃_n (f) and inverse theorem in a best approximation case such that both pervious theorems are in weighted space where the weight function is a generalized Jacobi (GL) weighted u(x). Finally we try to estimate degree of best one-sided approximation of the derivative of the function f in weighted space
Keywords: Hermite-Fejer, Interpolation, Polynomials, Weighted Spaces
