Some Best Constants in the Landau Inequality on a Finite Interval

Abstract

An additive form of the Landau inequality forf∈Wn∞[−1,1],‖f(m)‖⩽1cm1−mnT(m)n(1)‖f‖+cn−m2n−1n!mnT(m)n(1)‖f(n)‖is proved for 0<c⩽(cos(π/2n))−2, 1⩽m⩽n−1, with equality forf(x)=Tn(1+(x−1)/c), 1⩽c⩽(cos(π/2n))−2, whereTnis the Chebyshev polynomial. From this follows a sharp multiplicative inequality,‖f(m)‖⩽(2n−1n!)−m/nT(m)n(1)‖f|1−m/n‖f(n)‖m/nfor ‖f(n)‖⩾σ‖f‖, 2n−1n!cos2n(π/2n)⩽σ⩽2n−1n!, 1⩽m⩽n−1. For these values ofσ, the result confirms Karlin's conjecture on the Landau inequality for intermediate derivatives on a finite interval. For the proof of the additive inequality a Duffin and Schaeffer-type inequality for polynomials is shown.