Functions of Completely Regular Growth in the Half-Plane or a Cone

1992 ◽  
pp. 141-243 ◽  
Author(s):  
L. I. Ronkin
Keyword(s):  
Half Plane ◽  
Regular Growth ◽  
Completely Regular

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

2020 ◽  
Vol 30 (3) ◽  
pp. 396-409
Author(s):  
K.G. Malyutin ◽  
M.V. Kabanko
Keyword(s):  
Meromorphic Function ◽  
Half Plane ◽  
Meromorphic Functions ◽  
Growth Function ◽  
Regular Growth ◽  
Completely Regular ◽  
Upper Half Plane ◽  
Definition Of ◽  
Half Axis ◽  
Increasing Function

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 $p>1$.

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