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(z+n) for each n∈%TCIMACRO2124 %%BeginExpansionZ%EndExpansion. It is possible to obtain a function with certain characteristics for eachn∈%TCIMACRO2124 %%BeginExpansionZ%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,%\[fₙ( z) =\{ f( z+n) :n∈%TCIMACRO{{2124} }%%BeginExpansionZ%EndExpansion\} .\]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.