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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 0doi:10.2307/2043771 fatcat:yx7zntkzufd4lirh7o4ll3qjoq