SOME BEST CONSTANTS IN THE LANDAU-INEQUALITY ON A FINITE INTERVAL

An additive form of the Landau inequality for f is an element of W ''( infinity)[-1, 1], parallel to f((M))parallel to less than or equal to 1/c(m) (1 - m/n) T-n((m))(1) parallel to f parallel to + c(n-m)/2(n-1) n! m/n T-n((m)) (1) parallel to f((n))parallel to, is proved for 0<c l ess than or equal to(cos(pi/2n))(-2), 1 less than or equal to m less t han or equal to n-1, with equality for f(x) = T-n(1+(x-1)/c), 1 less t han or equal to c less than or equal to(cos(pi/2n))(-2), where T-n is the Chebyshev polynomial. From this follows a sharp multiplicative ine quality, parallel to f((m))parallel to less than or equal to (2(n-1)n! )(-m/n) T-n((m)) (1) parallel to f parallel to(1-m/n) parallel to f((n ))parallel to(m/n) for parallel to f((n))parallel to greater than or e qual to sigma parallel to f parallel to, 2(n-1)n! cos(2n)(pi/2n) less than or equal to sigma less than or equal to 2(n-1)n!, 1 less than or equal to m less than or equal to n-1. For these values of sigma, the r esult confirms Karlin's conjecture on the Landau inequality for interm ediate derivatives on a finite interval. For the proof of the additive inequality a Duffin and Schaeffer-type inequality for polynomials is shown. (C) 1998 Academic Press.