Analog of Hayman's Theorem and its Application to Some System of
Linear Partial Differential Equations

Andriy Bandura

Ivano-Frankivsk National Technical University of Oil and Gas, 15 Karpatska Str., Ivano-Frankivsk, 76019, Ukraine
E-mail: andriykopanytsia@gmail.com

Oleh Skaskiv

Ivan Franko National University of Lviv, 1 Universytetska Str., Lviv, 79000, Ukraine
E-mail: olskask@gmail.com

Received October 28, 2017, revised November 6, 2017.

Abstract

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 $l.$

Mathematics Subject Classification 2000: 32W50, 32A15, 32A22,
35G35, 32A40. Key words: entire function, bounded $\mathbf{L}$-index in joint variables, linear higher-order systems of PDE,
analytic theory of PDE, entire solution, linear higher-order differential equation.