Evaluation of the growth order of a class of entire functions on real axis

For a class of entire functions we study the question of the order of growth of functions on the real axis. This question is relevant to substantiate the integral representations of bounded solutions to some differential equations in partial derivatives, which were studied in other works of the authors. To evaluate the growth order of the function from this class we use the method of differential equations. The method consists, first, in the construction of a system of ordinary differential equations of the first order, the solution to which is a vector-function of trace function and its derivatives on the real axis. Secondly, under corresponding changes in the system of equations, we derive the rating estimate of the system of equations for large positive values of argument. The resulting estimate is non-trivial and shows how a complex parameter power series affects the order of growth of a function.