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International Journal of Mathematics and Mathematical Sciences
Volume 28, Issue 7, Pages 419-425

A generalization of Ky Fan's inequality

Department of Mathematics, University of Michigan, 2074 East Hall, 525 East University Avenue, Ann Arbor 48109, MI, USA

Received 20 March 2001; Revised 17 July 2001

Copyright © 2001 Hindawi Publishing Corporation. 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.


Let Pn,r(x) be the generalized weighted means. Let F(x) be a C1 function, y=y(x) an implicit decreasing function defined by f(x,y)=0 and 0<m<Mm, n2, xi[m,M], yi[m,M]. Then for 1r1, if fx/fy1, |(F(Pn,1(y))F(Pn,r(y)))/(F(Pn,1(x))F(Pn,r(x)))|<(maxmξM|F(ξ)|)/(minmηM|F(η)|)M/mM/m A similar result exists for fx/fy1. By specifying f(x,y) and F(x), we get various generalizations of Ky Fan's inequality. We also present some results on the comparison of Pn,sα(y)Pn,rα(y) and Pn,sα(x)Pn,rα(x) for sr, α.