An extension of some inequalities of P. Erdős and P. Turán concerning algebraic polynomials having all real zeros (Q1288915)

The authors consider inequalities of the following type \[ \int_{-1}^1 \rho(x) (P'_n(x))^pdx \leq K_n \int_{-1}^1 \rho(x) (P_n(x))^pdx \] where the weight function \(\rho:[-1,1]\to[0,\infty)\) and the power \(p>0\) are fixed and \(P_n\) is an arbitrary algebraic polynomial of degree at most \(n\) belonging to a prescribed class of polynomials. The problem is to find the constant \(K_n\) such that the above inequality is valid and sharp. The authors discuss the following cases in detail and obtain the best possible constant \(K_n\): (i) \(\rho(x)=(1-x^2)^\alpha\) (where \(\alpha>-1\)), \(p=2\), and \(P_n\in L_n\); (ii) \(\rho(x)=(1-x^2)^3\), \(p=4\), and \(P_n\in L_n\); (iii) \(\rho(x)=(1-x^2)^\alpha\) (where \(\alpha>-1\)), \(p=2\), and \(P_n\in H_n\); (iv) \(\rho(x)=(1-x^2)^3\), \(p=4\), and \(P_n\in H_n\); (v) \(\rho(x)=1\), \(p\geq 2\) even, and \(P_n\in H_n\), \noindent where \(L_n\) denotes the class of polynomials whose degree is \(n\), and \(P_n\in L_n\) is of the form \[ P_n(x)=\sum_{k=0}^na_k(1+x)^{n-k}(1-x)^k \qquad (a_k\geq 0), \] and \(H_n\) denotes the class of those polynomials whose degree is \(n\) and whose zeros are all real and lie inside \((-1,1)\).