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Let P,(x) be an algebraic polynomial of degree (n/2) f l IP2 (x) dx. The result is essentially best possible. Other related results are also proved. Let Hn be the set of all polynomials whose degree does not exceed n, i.e., polynomials of the form P (x) = cO + cIx + C2x2 + * + CnXn. Here the coefficients c0, c1, . . . , cn are arbitrary real numbers. The following inequalities on algebraic polynomials are well known. THEOREM A. Let Pn(x) E Hn; then we have (1.1) _lmax (1 X2)P,7(x) S n2 max p2

An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside [-1,+1]