International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 11, Pages 1781-1793
doi:10.1155/IJMMS.2005.1781

The case of equality in Landau's problem

G. W. Hagerty1 and P. Nag2

1Department of Mathematics, Black Hills State University, Spearfish 57799-9115, SD, USA
2Department of Mathematics, Black Hills State University, Spearfish 57799-9127, SD, USA

Received 11 January 2005

Copyright © 2005 G. W. Hagerty and P. Nag. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Kolmogorov (1949) determined the best possible constant Kn,m for the inequality Mm(f)≤Kn,mM0(n−m)/n(f)Mnm/n(f),  0<m<n, where f is any function with n bounded, piecewise continuous derivative on ℝ and Mk(f)=supx∈ℝ|f(k)(x)|. In this paper, we provide a relatively simple proof for the case of equality.