On two polynomial inequalities of Erdös related to those of the brothers Markov (Q1908224)

Let \(\widetilde\pi_n\) be the set of all algebraic polynomials \(p\) of degree at most \(n\) which can be expressed as \(p(x)= \sum^n_{k= 0} A_k(1+ x)^k (1- x)^{n- k}\), where \(A_k\geq 0\), \(k= 0,\dots, n\). The class \(\widetilde\pi_n\) contains, in particular, each polynomial of degree at most \(n\), which does not vanish in the open unit disk and takes positive values on \((- 1,1)\). The authors consider extensions of the Chebyshev inequality and some theorems of Erdös and Laguerre to polynomials \(p\in \widetilde\pi_n\): the exact pointwise bounds for \(|p(x^*)|\), \(|x^*|\geq 1\), and for \(|p'(x_0)|\), \(|x_0|\leq 1\), in terms of the uniform norm of \(p\) on \([- 1, 1]\) are found. Lower and upper estimates for some Maclaurin coefficients for \(p\in \widetilde\pi_n\) are also given.