Refined Growth Estimates for Composite Meromorphic and Entire Functions Using φ-order Framework

This work investigates the growth behaviour of composite meromorphic and entire functions through the concepts of φ-order and hyper φ-order, which offer a finer scale than the traditional and logarithmic orders. We derive inequalities that connect the growth of a composite function with that of its constituent functions. These findings provide a deeper insight into intermediate growth rates and present a practical framework for studying the composition of meromorphic and entire functions.