Some inequalities of algebraic polynomials

Abstract:

Erdös and Lorentz showed that by considering the special kind of the polynomials better bounds for the derivative are possible. Let us denote by ${H_n}$ the set of all polynomials whose degree is n and whose zeros are real and lie inside $[ - 1,1)$. Let ${P_n} \in {H_n}$ and ${P_n}(1) = 1$; then the object of Theorem 1 is to obtain the best lower bound of the expression $\smallint _{ - 1}^1|P_n’(x){|^p} dx$ for $p \geq 1$ and characterize the polynomial which achieves this lower bound. Next, we say that ${P_n} \in {S_n}[0,\infty )$ if ${P_n}$ is a polynomial whose degree is n and whose roots are all real and do not lie inside $[0,\infty )$. In Theorem 2, we shall prove Markov-type inequality for such a class of polynomials belonging to ${S_n}[0,\infty )$ in the weighted ${L_p}$ norm (p integer). Here ${\left \| {{P_n}} \right \|_{{L_p}}} = {(\smallint _0^\infty |{P_n}(x){|^p}{e^{ - x}} dx)^{1/p}}$. In Theorem 3 we shall consider another analogous problem as in Theorem 2.