Ž . JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 215, 461470 1997 ARTICLE NO. AY975645 Hadamard’s Inequality for r-ConvexFunctions P. M. Gill and C. E. M. Pearce Department of Applied Mathematics, The Uni  ersity of Adelaide, Adelaide, SA 5005, Australia and J. Pecaric ˇ ´ Faculty of Textile Technology, Uni  ersity of Zagreb, Pierottije  a 6, 11000, Zagreb, Croatia Submitted by A. M. Fink Received October 15, 1996 Versions of the upper Hadamard inequality are developed for r-convex and r-concave functions.  1997 Academic Press 1. INTRODUCTION   Hadamard’s inequality states that if f : a, b   is convex, then a  b 1 f a  f b Ž . Ž . b f  f t dt  . Ž. H ž / 2 b  a 2 a The first part of this result is subsumed under Jensen’s inequality and its many extensions. We shall be concerned with the second part, which is much less represented in the literature.   Recall that a positive function f is log-con  ex on a real interval a, b if     for all x, y  a, b and   0,1 we have  1 f  x  1   y  f x f y . 1.1 Ž . Ž . Ž . Ž . Ž . If the reverse inequality holds, f is termed log-conca  e. 461 0022-247X97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.