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Gruss Type Discrete Inequalities in Inner Product Spaces, Revisited

Sever Silvestru Dragomir

TL;DR

This paper revisits Grüss-type inequalities in real and complex inner product spaces, deriving sharp bounds for weighted vector sequences and establishing a midpoint-positivity lemma that connects positivity conditions to a bound about deviations from the midpoint. It elaborates a general Grüss-type framework, including a forward-difference variant and a bound for the difference between weighted inner products and the inner product of weighted sums, with best-possible constants such as $1/2$ and $1/4$. The results are then applied to convex, Fréchet-differentiable functions to obtain a reverse Jensen inequality, giving explicit upper bounds in terms of gradient spread and data dispersion, along with corollaries and special cases. Overall, the work extends Grüss-type inequalities and provides practical, sharp bounds for deviations from Jensen-type inequalities in Hilbert-space settings.

Abstract

Some sharp inequalities of Gruss type for sequences of vectors in real or complex inner product spaces are obtained. Applications for Jensen's inequality for convex functions defined on such spaces are also provided.

Gruss Type Discrete Inequalities in Inner Product Spaces, Revisited

TL;DR

This paper revisits Grüss-type inequalities in real and complex inner product spaces, deriving sharp bounds for weighted vector sequences and establishing a midpoint-positivity lemma that connects positivity conditions to a bound about deviations from the midpoint. It elaborates a general Grüss-type framework, including a forward-difference variant and a bound for the difference between weighted inner products and the inner product of weighted sums, with best-possible constants such as and . The results are then applied to convex, Fréchet-differentiable functions to obtain a reverse Jensen inequality, giving explicit upper bounds in terms of gradient spread and data dispersion, along with corollaries and special cases. Overall, the work extends Grüss-type inequalities and provides practical, sharp bounds for deviations from Jensen-type inequalities in Hilbert-space settings.

Abstract

Some sharp inequalities of Gruss type for sequences of vectors in real or complex inner product spaces are obtained. Applications for Jensen's inequality for convex functions defined on such spaces are also provided.

Paper Structure

This paper contains 2 sections, 9 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Some Grüss Type Inequalities

Key Result

Theorem 1

Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right)$ be an inner product space over the real or complex number field $\mathbb{K}$$\left( \mathbb{K=R},\mathbb{C}\right) ,$$\overline{\mathbf{x}}=\left( x_{1},\dots ,x_{n}\right) \in H^{n}$, $\overline{\mathbf{\alpha }}=\left( \alpha _{1},\dot then we have the inequality The constant $\frac{1}{4}$ is the best possible one in the sense that i

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 4
  • proof
  • Remark 2
  • ...and 8 more