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Improved eigenvalue inequalities via two major subclasses of superquadratic functions

Mohsen Kian

Abstract

There exist two major subclasses in the class of superquadratic functions, one comprises concave and decreasing functions, while the other consists of convex and monotone increasing functions. Leveraging this distinction, we introduce eigenvalue inequalities for each case. The characteristics of these functions allow us to advance our findings in two ways: firstly, by refining existing results related to eigenvalues for convex functions, and secondly, by deriving complementary inequalities for other function types. To bolster our claims, we will provide illustrative examples.

Improved eigenvalue inequalities via two major subclasses of superquadratic functions

Abstract

There exist two major subclasses in the class of superquadratic functions, one comprises concave and decreasing functions, while the other consists of convex and monotone increasing functions. Leveraging this distinction, we introduce eigenvalue inequalities for each case. The characteristics of these functions allow us to advance our findings in two ways: firstly, by refining existing results related to eigenvalues for convex functions, and secondly, by deriving complementary inequalities for other function types. To bolster our claims, we will provide illustrative examples.
Paper Structure (3 sections, 13 theorems, 81 equations, 1 figure)

This paper contains 3 sections, 13 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Result

Key Result

Lemma 1.1

ajsabm If $f:[0,\infty)\to\mathds{R}$ is superquadratic, then

Figures (1)

  • Figure 1: Graphs of some functions in the two subclasses of superquadratic functions

Theorems & Definitions (24)

  • Lemma 1.1
  • Lemma 1.2: The Minimax Principle
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • ...and 14 more