Caution: Heavy citing
"In his doctoral dissertation Revankar (1967) expounded his generalized
production functions that permit variability to returns-to-scale as well as elasticity of substitution. In contrast with the production functions that (rather unrealistically) assume the same returns to scale at all levels of output, Zellner and Revankar (1969) found a procedure to generalize any given (neoclassical) production function with specified constant or variable elasticities of substitution such that the resulting production function retains its specification as to the elasticities of substitution all along but permits returns- to-scale to vary with the scale of output. Their Generalized Production Function (GPF) is given as
\begin{equation}
\ Pe^{\theta P} = c^h f^h
\end{equation}
where f is the basic function (e.g. Cobb-Douglas, CES, etc) as the
object of generalization, c is the constant of integration and θ, h
relate to parameters
associated with the returns-to-scale function. In particular, if the Cobb-Douglas production function is generalized, we have
\begin{equation}
\ Pe ^{\theta P}= AK^{\rho\alpha} L^ {\rho (1- \alpha )}
\end{equation}. This function is interesting
from the viewpoint of estimation also. It has to be estimated so as to maximize the likelihood function since the Least Squares and Max Likelihood estimators of parameters do not coincide. The return to scale function is given by
\begin{equation}
\rho (P) = \rho /(1 + \theta P)
\end{equation} . Depending on θ the sign of θ , the returns-to-scale function monotonically increases or decreases with
increase in P. However, as we know, the returns to scale first increases with output, remains more or less constant in a domain and then begins falling. This fact is not captured by the Zellner-Revankar function since it gives us a linear returns-to-scale function."
from A Brief History of Production Functions by SK Mishra 2007
So, I would guess, the \begin{equation} \alpha \end{equation} parameter
in your log-linearized function would have something to do with returns-to-scale.
Regarding estimation, I'm way too much of a newb. But this might be something at least similar to what you are looking for: https://link.springer.com/chapter/10.1007/3-540-28556-3_2
Variable Elasticity of Substitution and Economic
Growth: Theory and Evidence by
Giannis Karagiannis,
Theodore Palivos,
and Chris Papageorgiou,
2005
