Under some simple conditions on real function f defined on an interval I, the bivariable functions given by the following formulas Afx,y:=fx+y-fy,Gfx,y:=fxfyy,andHfx,y:=xyfx+y-fy,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} A_{f}\\left( x,y\\right):= & {} f\\left( x\\right) +y-f\\left( y\\right) , \\\\ G_{f}\\left( x,y\\right):= & {} \\frac{f\\left( x\\right) }{f\\left( y\\right) }\\,y, \\\\ \\text{ and } \\quad H_{f}\\left( x,y\\right):= & {} \\frac{xy}{f\\left( x\\right) +y-f\\left( y\\right) }, \\end{aligned}$$\\end{document}for all x,yin I, generalize, respectively, the classical weighted arithmetic, geometric and harmonic means. The invariance equations Af∘Gg,Hh=Af,Gg∘Af,Hh=GgandHh∘Af,Gg=Hh,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} A_{f}\\circ \\left( G_{g},H_{h}\\right) =A_{f}, \\quad G_{g}\\circ \\left( A_{f},H_{h}\\right) =G_{g} \\quad \\text{ and } \\quad H_{h}\\circ \\left( A_{f},G_{g}\\right) =H_{h}, \\end{aligned}$$\\end{document}where f, g, h are the unknown functions are, in some special cases, solved. The convergence of iterates of the relevant mean-type mappings is considered. As an application the solutions of some functional equations are determined.