Multivariate Mond-Pecaric Method with Applications to Hypercomplex Function Sobolev Embedding

TL;DR

This paper extends the Mond-Pecaric method to multivariate hypercomplex functions, deriving new inequalities and applying them to Sobolev embedding theorems for hypercomplex functions with operator inputs.

Contribution

It introduces a multivariate extension of the Mond-Pecaric method and applies it to derive inequalities and Sobolev embeddings for hypercomplex functions.

Findings

Derived fundamental inequalities for multivariate hypercomplex functions.

Established Sobolev embedding results for hypercomplex functions with operators.

Presented ratio and difference inequalities for these functions.

Abstract

Mond and Pecaric introduced a method to simplify the determination of complementary inequalities for Jensen's inequality by converting it into a single-variable maximization or minimization problem of continuous functions. This principle has significantly enriched the field of operator inequalities. Our contribution lies in extending the Mond-Pecaric method from single-input operators to multiple-input operators. We commence by defining normalized positive linear maps, accompanied by illustrative examples. Subsequently, we employ the Mond-Pecaric method to derive fundamental inequalities for multivariate hypercomplex functions bounded by linear functions. These foundational inequalities serve as the basis for establishing several multivariate hypercomplex function inequalities, focusing on ratio relationships. Additionally, we present similar results based on difference relationships. Finally, we apply the derived multivariate hypercomplex function inequalities to establish Sobolev embedding via Sobolev inequality for hypercomplex functions with operator inputs.