Invariance problems of generalized weighted classical means

Under certain simple conditions for real functions f, g, h, defined on a real interval, the bivariable functions Af , Gg and Hₕ given, respectively, by $${aligned} Af( x,y)= & f( x) +y-f( y) , Gg( x,y) =g( x) /g( y) y,\\ Hₕ( x,y)= & xy/x-h( x) +h( y) , {aligned}$$ are natural generalizations of the classical weighted arithmetic, geometric and harmonic means. The article concerns the following invariance equations involving these means $${aligned} Af ∘ ( Ag,Aₕ) =Af, Gf ∘ ( Gg,Gₕ) =Gf, Hf ∘ ( Hg,Hₕ) =Hf, {aligned}$$ where f, g and h are unknown functions. The first two of these equations are investigated under the assumption that f is twice differentiable, and g, h are differentiable. If $$Af$$ is translative and $$Gf$$ and $$Hf$$ are homogeneous, we determine the solutions without any regularity conditions.