Mean value theorems for solutions of linear partial differential equations

We consider generalized mean value theorems for solutions of linear differential equations with constant coefficients and zero right-hand side which satisfy the following homogeneity condition with respect to a given vector $\mathbf M$ with positive integer components: for each partial derivative occurring in the equation, the inner product of the vector composed of the orders of this derivative in each variable by the vector $\mathbf M$ is independent of the derivative. The main results of this paper generalize the well-known Zalcman theorem. Some corollaries are given.