On the invariance equation for means of generalized power growth

We generalize the result of (Witkowski, 2014) which binds orders of homogeneous, symmetric means $M,N,K \colon\mathbb{R}_+^2 \to \mathbb{R}_+$ of power growth that satisfy the invariance equation $K(M(x,y),N(x,y))=K(x,y)$ to the broader class of means. Moreover, we define the lower- and the upper-order which gives us insight into the order of the solution of this equation in the case when means do not belong to this class.