The meromorphic functions of completely regular growth on the upper half-plane

Abstract

A strictly positive continuous unbounded increasing function $\gamma(r)$ on the half-axis $[0,+\infty)$ is called growth function. Let the growth function $\gamma(r)$ satisfies the condition $\gamma(2r)\leq M\gamma(r)$ for some $M>0$ and for all $r>0$. In the paper, the class $JM(\gamma(r))^o$ of meromorphic functions of completely regular growth on the upper half-plane with respect to the growth function $\gamma$ is considered. The criterion for the meromorphic function $f$ to belong to the space $JM(\gamma(r))^o$ is obtained. The definition of the indicator of function from the space $JM(\gamma(r))^o$ is introduced. It is proved that the indicator belongs to the space $\mathbf{L}^p[0,\pi]$ for all $p>1$.

Asymptotics of Zeros of an Entire Function With an Integral Representation, Connected by a Regular Loaded Differentiation Operator on an Interval

“…Meromorphic functions of completely regular growth in the upper half-plane with respect to the growth function are studied in [10]. The issues of the location of zeros of an entire function: on one ray, on a straight line, on several rays, in an angle or arbitrarily in the complex plane have been studied in numerous works [1], [11] [19].…”