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 $$A_{f}$$ , $$G_{g}$$ and $$H_{h}$$ given, respectively, by $$\begin{aligned} A_{f}\left( x,y\right)= & f\left( x\right) +y-f\left( y\right) , \qquad G_{g}\left( x,y\right) =\frac{g\left( x\right) }{g\left( y\right) }y,\\ H_{h}\left( x,y\right)= & \frac{xy}{x-h\left( x\right) +h\left( y\right) }, \end{aligned}$$ are natural generalizations of the classical weighted arithmetic, geometric and harmonic means. The article concerns the following invariance equations involving these means $$\begin{aligned} A_{f} \circ \left( A_{g},A_{h}\right) =A_{f}, \quad G_{f} \circ \left( G_{g},G_{h}\right) =G_{f}, \quad H_{f} \circ \left( H_{g},H_{h}\right) =H_{f}, \end{aligned}$$ 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 $$A_{f}$$ is translative and $$G_{f}$$ and $$H_{f}$$ are homogeneous, we determine the solutions without any regularity conditions.