On a Problem of Turan About Polynomials with Curved Majorants

Q. I. Rahman
1972 Transactions of the American Mathematical Society  
Let (x) ä 0 for -1 £j * S 1. For a fixed x0 in [ -1, 1 ] what can be said for max |pi(*o)l if pn(x) belongs to the class P0 of all polynomials of degree n satisfying the inequality \p"(x)\?=(x) for -lgxgl? The case (x)=l was considered by A. A. Markov and S. N. Bernstein. We investigate the problem when (j>(x) = (l -x2)1'2. We also study the case (x) = \x\ and the subclass consisting of polynomials typically real in |z| < 1. Received by the editors February 1, 1971. AMS 1970 subject
more » ... subject classifications. Primary 30A06, 30A40; Secondary 42A04. Key words and phrases. Polynomials with curved majorants, Chebyshev polynomial of the first kind, Chebyshev polynomial of the second kind, polynomials typically real in \z\ < 1, Gauss-Lucas theorem.
doi:10.2307/1995732 fatcat:pd4rdh6on5c7lo4eqpv7ytkdxm