### Some Inequalities Related to Holder's Inequality

C. J. Neugebauer
1981 Proceedings of the American Mathematical Society
In this paper we generalize the Jodeit-Jones-Moser inequalities relating 0<*-(jS/r°,if U/H, < 1. 1. Let (X, ?flc, fi) be a measure space, and let f,: X -» [0, oo] be measurable. Throughout the paper 1 < q < oo, \/q + 1/p = 1, and \\f\\q < 1. Also let there be given an indexing map o: [0, oo) -» 9H such that a(r') c o(r"), r' < r", and a(0) = 0. We will write o(r) = Sr and we will assume that H^H^, < oo, where r = -Xsr-Throughout the paper, 1 < a < p unless q = 1, in which case 1 < a < oo, and 0
more » ... e 1 < a < oo, and 0 < ß < a. We set ftr) = ||: [0, oo) -* [0, oo) be nonincreasing, and let m* be the measure on [0, oo) induced by the nondecreasing function \¡/*. If\f>* is continuous, then f °° *{^*(r) -F*(r)} dm* 2, i>(«) = e_u is due to Moser , and the case q > 1 with 1, and the same special X, a, as above, the inequahty is due to Jones  . The proof in  is different from those in  and  and, as we shall see, lends itself to prove our theorem.