SOME INEQUALITIES OF ALGEBRAIC POLYNOMIALS

Erdos and Lorentz showed that by considering the special kind of the p olynomials better bounds for the derivative are possible. Let us denot e by H-n the set of all polynomials whose degree is n and whose zeros are real and lie inside [-1, 1). Let P-n is an element of H-n and P-n( 1) = 1; then the object of Theorem 1 is to obtain the best lower bound of the expression integral(-1)(1)\P'(n)(x)\(p) dx for p greater than or equal to 1 and characterize the polynomial which achieves this lowe r bound. Next, we say that P-n is an element of S-n[0, infinity). if P -n is a polynomial whose degree is n and whose roots are all real and do not lie inside [0, infinity). In Theorem 2, we shall prove Markov-t ype inequality for such a class of polynomials belonging to S-n[0, inf inity) in the weighted L(p) norm (p integer). Here parallel to P-n par allel to(Lp) = (integral(0)(infinity)\P-n(x)\(p)e(-x) dX)(1/P). In The orem 3 we shall consider another analogous problem as in Theorem 2.