In this paper, we will prove some fundamental properties of the power mean operator Mpg(t)=(1ϒ(t)∫0tλ(s)gp(s)ds)1/p,for t∈I⊆R+,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M}_{p}g(t)= \\biggl( \\frac{1}{\\Upsilon(t)} \\int _{0}^{t} \\lambda (s)g^{p} ( s ) \\,ds \\biggr) ^{1/p},\\quad\ ext{for }t\\in \\mathbb{I}\\subseteq \\mathbb{R}_{+}, $$\\end{document}of order p and establish some lower and upper bounds of the compositions of operators of different powers, where g, λ are a nonnegative real valued functions defined on mathbb{I} and Upsilon(t)=int _{0}^{t}lambda ( s ) ,ds. Next, we will study the structure of the generalized class mathcal{U}_{p}^{q}(B) of weights that satisfy the reverse Hölder inequality Mqu≤BMpu,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M}_{q}u\\leq B\\mathcal{M}_{p}u, $$\\end{document}for some p< q, p.qneq 0, and B>1 is a constant. For applications, we will prove some self-improving properties of weights in the class mathcal{U}_{p}^{q}(B) and derive the self improving properties of the weighted Muckenhoupt and Gehring classes.