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Some Generalizations of Mercer inequality and its operator extensions

Mohsen Kian, Zainab Peymani Mazraj

Abstract

We study the Mercer inequality and its operator extension for superquadratic functions. In particular, we give a more general form of the Mercer inequality by replacing some constants by positive operators. As some consequences, our results produce a Jensen operator inequality for superquadratic functions. Moreover, we present some Mercer inequalities of Hermite-Hadamard's type.

Some Generalizations of Mercer inequality and its operator extensions

Abstract

We study the Mercer inequality and its operator extension for superquadratic functions. In particular, we give a more general form of the Mercer inequality by replacing some constants by positive operators. As some consequences, our results produce a Jensen operator inequality for superquadratic functions. Moreover, we present some Mercer inequalities of Hermite-Hadamard's type.
Paper Structure (2 sections, 10 theorems, 62 equations)

This paper contains 2 sections, 10 theorems, 62 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Main Result

Key Result

Lemma 2.1

Let $0\leq m < M$ and let $f : [0,\infty)\to \mathbb{R}$ be a superquadratic function. Then for all $x, y \in [m, M]$ and every $\lambda\in[0,1]$.

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 8 more