On the growth rates of the composition of integer translated entire and meromorphic functions

Manab Biswas; Debashis Kumar Mandal

doi:10.5644/SJM.21.01.14

Authors

Manab Biswas Department of Mathematics, Kalimpong College, Kalimpong, Dist- Kalimpong, PIN-734301, West Bengal, India
Debashis Kumar Mandal Department of Mathematics, Cooch Behar College, Cooch Behar, Dist-Cooch Behar, PIN-736101, West Bengal, India

DOI:

https://doi.org/10.5644/SJM.21.01.14

Keywords:

Entire function, meromorphic function, $\left( {p,q}\right) $-order and lower $\left( {p,q}\right) $-order, integer translation, growth

Abstract

The integer translation of a function $f(z)$ is denoted by $f\left(
z+n\right) $ for each $n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
.$ It is possible to obtain a function with certain characteristics for each
$n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
.$ This study examines the impact of integer translations on the growth and behavior of a meromorphic function $f(z)$. Specifically, we consider the family of meromorphic functions generated by integer shifts of $f(z)$, denoted as,%
\[
f_{n}\left( z\right) =\left\{ f\left( z+n\right) :n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\right\} .
\]
The primary focus is on understanding how these integer translations affect the Nevanlinna characteristic function $T(r,f),$ which is a key tool for assessing the growth of meromorphic functions. Our study also includes a comparative growth analysis between integer-translated versions of both entire and meromorphic functions. By examining a range of conditions, we provide insights into how translation influences the growth and value distribution of the functions. This investigation contributes to a deeper understanding of translation-invariant properties in complex analysis and offers new perspectives on the dynamic growth behavior of meromorphic and entire functions.

Statistics

Abstract: 220 / PDF: 66

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