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

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.