A note on a result of Zolotarev and Bernstein

In this note we obtain error bounds to x²ⁿ⁺¹–x²ⁿ on [–1, 1] by polynomials of degree at most (2n–1). The result proved here improves and extends some of the known results of Zolotarev and Bernstein. The proof presented here is different (and simple) from the one adopted by Zolotarev and Bernstein.