An Inductive Rearrangement Theorem

Abstract

In [2, Theorem 3.2, p. 429] and [3, Theorem 2.1, p. 155], the author established two induction theorems which gave rise to a series of fundamental results in spectral and rearrangement inequalities. In particular, the classical inequalities of Hardy-Littlewood-Pólya [4, Theorem 108, p. 89] and Pólya [6] were derived and conditions for equalities obtained (see [2, Theorem 3.8, p. 433] and [3, Theorem 2.6, p. 157]). In [5, Theorem 6, p. 651 ; and Theorem 20, p. 569], Fischer and Holbrook also gave alternative conditions for equalities to hold in the aforesaid inequalities. In this paper, we show that the result of Fischer and Holbrook can be proved by induction using an inductive rearrangement theorem which turns out to be a stronger version of the induction theorems given in [2, Theorem 3.2, p. 429] and [3, Theorem 2.1, p. 155].