This paper investigates the distribution function and nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\)-valued functions equipped with the hyperbolic norm. It begins by introducing the concept of the distribution function for \( \mathbb{B}\mathbb{C}\)-valued functions, which characterizes valuable insights into the behavior and structure of \(\mathbb{B}\mathbb{C}\)-valued functions, allowing to analyze their properties and establish connections with other mathematical concepts. Next, the nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\)-valued functions with the hyperbolic norm are studied. By exploring the nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\)-valued functions, it is aimed to determine how the hyperbolic norm influences the rearrangement process and its impact on the function’s behavior and properties.

本文研究了配备双曲范数的\(\mathbb{B}\mathbb{C}\)值函数的分布函数和非增重排。首先介绍了\( \mathbb{B}\mathbb{C}\)值函数的分布函数的概念，它描述了对\(\mathbb{B}\mathbb{C}的行为和结构的有价值的见解}\)值函数，允许分析它们的属性并与其他数学概念建立联系。接下来，研究具有双曲范数的\(\mathbb{B}\mathbb{C}\)值函数的非增重排。通过探索\(\mathbb{B}\mathbb{C}\)值函数的非增重排，旨在确定双曲范数如何影响重排过程及其对函数行为和性质的影响。