Landau’s inequality for the difference operator

Abstract:

The best constants for Landau’s inequality with the classical $p$-norms are known explicitly only when $p = 1,2{\text { and }}\infty$. This is true for both the discrete and the continuous versions of the inequality and for both the "whole line" and "half line" cases. In each of the six known cases the best constant in the discrete version is the same as the best constant for the continuous version. Here we show that for many other values of $p$ the discrete constants are strictly greater than the corresponding continuous ones. In addition, we show that the "three norm version" of the inequality, established by Nirenberg and Gabushin in the continuous case is also valid in the discrete case.