An extremal problem for real algebraic polynomials (Q485555)

Let \(\mathcal{P}_{n}\) be the set of all real algebraic polynomials of degree at most \(n\) which are positive on the interval \((-1,1)\) and have no zeros inside the unit circle (\(| z| <1\)). The main result is as follows. Suppose that \(\alpha\) and \(\beta\) are real numbers with \(q=\max \{\alpha,\beta\}\geq-1/2\), \(r\) is a positive integer and \(P_{n}\) is a polynomial belonging to \(\mathcal{P}_{n}.\) Then \(\max_{-1\leq x\leq1}(P_{n}(x))^{r}\) is bounded above by \[ C(n,r,\alpha,\beta) \int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}(P_{n}(x))^{r}dx, \] where \(C(n,r, \alpha,\beta) =\frac{\Gamma(rn+\alpha+\beta+2)}{2^{\alpha+\beta +1} \Gamma (rn+s+1) \Gamma(q+1)}\) and \(s=\min\{\alpha,\beta\}\). Moreover, the constant \(C(n,r,\alpha,\beta)\) is optimal. For \(\alpha=\beta=0\) and \(r=1,\) this covers a result due to \textit{B. Sendov} [``An integral inequality for algebraic polynomials with only real zeros'', Annuaire Univ. Sofia Fac. Sci. Phys. Math. 53, 19--32 (1958/1959)].