International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 2, Pages 397-398

Note on Hölder inequalities

Sung Guen Kim

Department of Mathematics, Pohang Institute of Science & Technology, P.O. Box 215, Pohang, 790-600, Korea

Received 13 April 1992; Revised 26 June 1993

Copyright © 1995 Sung Guen Kim. 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.


In this note, we show that if m,n are positive integers and xij0, for i=1,,n, for j=1,,m, then (i=1nxi1xim)m(i=1nxi1m)(i=1nximm) with equality, in case (x11,,xn1)0 if and only if each vector (x1j,,xnj), j=1,,m, is a scalar multiple of (x11,,xn1). The proof is a straight-forward application of Hölder inequalities Conversely, we show that Hölder inequalities can be derived from the above result.