Some inequalities of algebraic polynomials with nonnegative coefficients

Abstract

Let S" be the collection of all algebraic polynomials of degree < n with nonnegative coefficients. In this paper we discuss the extremal problemis a positive and integrable function. This problem is solved completely in the cases (i) [a,b] = [-l, 1], ca(jc) = (1 -x2)" , a>-l; (ii) [a, b) = [0, co), a>(x) = xae~x , a > -1 ; (iii) (a, b) = (-co, oo), (o(x) = e~ax , a>0.The second case was solved by Varma for some values of a and by Milovanovic completely. We provide a new proof here in this case.