On the Soft Orders and Growth Properties of Entire Functions

This paper develops a formal mathematical framework for the analysis of parameterized collections of entire functions and their classical growth orders using the language of soft set theory. We strictly follow the paradigm where a soft set is a parameterized family of sets and a soft element is a specific, valid selection from that family. We introduce and formalize the soft order and soft lower order of a soft entire element, which associate each parameter with the classical growth order of its corresponding function. The work is structured into four main sections of results. The first section establishes the fundamental algebraic properties of the soft order under operations like sum and product. The second section investigates the behavior of the soft order under key analytic operations, presenting a detailed proof of the invariance of the soft order under differentiation. The third section develops a theory of soft relative order, proving analogues of classical comparison theorems. The final section delves into the relationship between the soft order and the intrinsic properties of the functions, providing substantial, detailed proofs for the formula connecting the soft order to Taylor coefficients and for the fundamental inequality relating the soft order to the distribution of zeros. Each section provides rigorous definitions, theorems, and proofs, creating a comprehensive and self-contained foundation for the study of parameterized growth properties in complex analysis.