Meromorphic functions of completely regular growth and their logarithmic derivatives

1992 ◽  
Vol 33 (1) ◽  
pp. 34-40 ◽  
Author(s):  
A. A. Gol'dberg ◽  
M. L. Sodin ◽  
N. N. Strochik
Keyword(s):  
Meromorphic Functions ◽  
Regular Growth ◽  
Completely Regular ◽  
Logarithmic Derivatives

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 $p>1$.

scholarly journals Logarithmic derivative estimates of meromorphic functions of finite order in the half-plane

2020 ◽  
Vol 54 (2) ◽  
pp. 172-187
Author(s):  
I.E. Chyzhykov ◽  
A.Z. Mokhon'ko
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Half Plane ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Sharp Estimates ◽  
Exceptional Sets ◽  
Upper Half Plane ◽  
Logarithmic Derivatives

We established new sharp estimates outside exceptional sets for of the logarithmic derivatives $\frac{d^ {k} \log f(z)}{dz^k}$ and its generalization $\frac{f^{(k)}(z)}{f^{(j)}(z)}$, where $f$ is a meromorphic function $f$ in the upper half-plane, $k>j\ge0$ are integers. These estimates improve known estimates due to the second author in the class of meromorphic functions of finite order.Examples show that size of exceptional sets are best possible in some sense.

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