Some Inequalities of Algebraic Polynomials

Erdös and Lorentz showed that by considering the special kind of the polynomials better bounds for the derivative are possible. Let us denote by Hn the set of all polynomials whose degree is n and whose zeros are real and lie inside [-1, 1 ). Let Pn € H„ and Pn ( 1 ) = 1 ; then the object of Theorem 1 is to obtain the best lower bound of the expression f}_{ \P¡,(x)\p dx for p > 1 and characterize the polynomial which achieves this lower bound. Next, we say that Pn £ Sn[0, oo) if Pn is a polynomial whose degree is n and whose roots are all real and do not lie inside [0, oo). In Theorem 2, we shall prove Markovtype inequality for such a class of polynomials belonging to S„[0, oo) in the weighted Lp norm (p integer). Here \\Pn\\Lp = (/0°° \Pn{x)\pe-xdx)1/? . In Theorem 3 we shall consider another analogous problem as in Theorem 2. Introduction Let Hn be the set of all polynomials whose degree is n and whose zeros are real and lie inside [-1, 1). Concerning this class of polynomials belonging to Hn we shall prove the following theorem. Theorem 1. Let P„ £ H„, subject to the condition P„(l) I. Then we have (for P>\) (i.i) /' \K(x)\ np pdx> 2P~l((n-l)p+l)' with equality iff Pn(x) = (LjL)" ■ The case p — 2 was considered in [5] and [8]. In 1964 G. Szegö [6] studied the order of magnitude of ||-f,íll¿cx>/ll-'«ll¿oo f°r unrestricted polynomials P„ of degree < n for the norm ||/>„|| = sup | J>„ (*)<>-* | *>o on (0, oo ). More precisely, he proved the following Received by the editors April 30, 1993 and, in revised form, July 6, 1993 and August 10, 1993. 1991 Mathematics Subject Classification. Primary 26D15. The author is deceased (December 8, 1994). ©1995 American Mathematical Society