Generalized Multivariate Hypercomplex Function Inequalities and Their Applications

TL;DR

This paper extends the Mond-Pecaric method to multivariate hypercomplex functions, using sigmoid-based bounds for improved inequalities and introducing the concept of W-boundedness for hypercomplex functions.

Contribution

It introduces a novel approximation approach for multivariate hypercomplex functions using sigmoid functions, expanding the scope of inequalities and bounds in this domain.

Findings

Derived new inequalities for multivariate hypercomplex functions.

Introduced W-boundedness as a generalization of R-boundedness.

Developed an approximation theory and bounds algebra for multivariate hypercomplex functions.

Abstract

This work extends the Mond-Pecaric method to functions with multiple operators as arguments by providing arbitrarily close approximations of the original functions. Instead of using linear functions to establish lower and upper bounds for multivariate functions as in prior work, we apply sigmoid functions to achieve these bounds with any specified error threshold based on the multivariate function approximation method proposed by Cybenko. This approach allows us to derive fundamental inequalities for multivariate hypercomplex functions, leading to new inequalities based on ratio and difference kinds. For applications about these new derived inequalities for multivariate hypercomplex functions, we first introduce a new concept called W-boundedness for hypercomplex functions by applying ratio kind multivariate hypercomplex inequalities. W-boundedness generalizes R-boundedness for norm mappings with input from Banach space. Additionally, we develop an approximation theory for multivariate hypercomplex functions and establish bounds algebra, including operator bounds and tail bounds algebra for multivariate random tensors.