Let mu be the Haar measure of a unimodular locally compact group G and m(G) as the infimum of the volumes of all open subgroups of G. The main result of this paper is that ∫Gf∘ϕ1∗ϕ2gdg≤∫Rf∘ϕ1∗∗ϕ2∗xdx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _{G}^{} f \\circ \\left( \\phi _1 * \\phi _2 \\right) \\left( g \\right) dg \\le \\int _{\\mathbb {R}}^{} f \\circ \\left( \\phi _1^* * \\phi _2^* \\right) \\left( x \\right) dx \\end{aligned}$$\\end{document}holds for any measurable functions phi _1, phi _2 :G rightarrow mathbb {R}_{ge 0} with mu ( textrm{supp} ; phi _1 ) + mu ( textrm{supp} ; phi _2 ) le m(G) and any convex function f :mathbb {R}_{ge 0} rightarrow mathbb {R} with f(0) = 0. Here phi ^* is the rearrangement of phi . Let Y_O(P,G) and Y_R(P,G) denote the optimal constants of Young’s and the reverse Young’s inequality, respectively, under the assumption mu ( textrm{supp} ; phi _1 ) + mu ( textrm{supp} ; phi _2 ) le m(G). Then we have Y_O(P,G) le Y_O(P,mathbb {R}) and Y_R(P,G) ge Y_R(P,mathbb {R}) as a corollary. Thus, we obtain that m (G) = infty if and only if H (p,G) le H (p, mathbb {R}) in the case of p' := p/(p-1) in 2 mathbb {Z}, where H(p, G) is the optimal constant of the Hausdorff–Young inequality.