The Miles | Shia Extremal Problem for Meromorphic Functions Determined by the Model Growth Function

The connection between the theory of Fourier series and complex analysis has been known for over a hundred years, since a power series considered on a circle is represented by a trigonometric series. The study of the connection between the boundary behavior of analytic and subharmonic functions, on the one hand, and Fourier series, on the other, led to profound results in both theories. Beginning in the 1960s, the American mathematicians L. Rubel and B. Taylor began to use a method for studying the asymptotic behavior of entire and meromorphic functions based on the Fourier series for the logarithm of the modulus of a meromorphic function. One of the advantages of this method is that it allows one to study functions with irregular growth at infinity and functions of infinite order. In addition, since the Fourier coeficients are expressed through the zeros and poles of a meromorphic function, they can be used to study the distribution of zeros and poles. One of the directions of this theory is finding the best upper and lower bounds for the upper and lower limits of the ratios of Nevanlinna characteristics. Such estimates were obtained at the end of the last century in a joint work by Miles and Shay. In the present paper, we extend some results from the work of Miles and Shay to classes of meromorphic functions defined by a model growth function.