We prove sharp inequalities between L-P-norms (1 < p < infinity) of functions Hf and H*f, where H is the Hardy operator, H* is its dual, and f is a nonnegative nonincreasing function on (0, infinity). In particular, we extend one result obtained for integer p >= 2 by Boza and Soria (2019), to the whole range of values p >= 2. (C) 2019 Elsevier Inc. All rights reserved.