Analog of Hayman's theorem and its application to some system of linear partial differential equations

We used the analog of known Hayman's theorem to study the boundedness of $\mathbf{L}$-index in joint variables of entire solutions of some linear higher-order systems of PDE's and found sufficient conditions providing the boundedness, where $\mathbf{L}(z)=(l_1(z), \ldots, l_{n}(z)),$ $l_j:\mathbb{C}^n\to \mathbb{R}_+$ is a continuous function $j\in\{1,\ldots,n\}.$ Growth estimates of these solutions are also obtained. We proposed the examples of systems of PDE's which prove the exactness of these estimates for entire solutions. The obtained results are new even for the one-dimensional case because of the weakened restrictions imposed on the positive continuous function $l.$