An Extremu M Problem concerning Algebraic Polynomials

Let S " he the set of all polynomials whose degree does not exceed n and whose all zeros are real but lie outside (-1, 1). Similarly, we say p"E Q " if p,(x) is a real polynomial whose all zeros lie outside the open disk with center at the origin and radius l. Further we will denote by H " the set of all polynomials of degree-n and of the form k=O where q,,k (x) = (1 +x)k (l-x)"-'. Elements of H,, are called polynoitiials with positive coefficients (in 1-x and 1 + .x) by G. G. Lorentz. The following inequalities for derivatives of polynomials of special type are known THEOREM A (P. Erdős). Let p " ES,; then max p,, (x)I-1 en max l p " (x)I .-1 r-, ? t-_i Further, tiic constant-h e can not he replaced by a smaller one. a Constant C r for which (1 .2) niax !p,') (X)! _ Cat max p " (x)j. 1)(2n+3) (1 .5) (1-.x)(p ;,(x)) d.x n(n+ (l-x)p 2 (x) dx 4(2n+1) _,