# Some Gruss type inequalities for Frechet differentiable mappings

## TL;DR

This paper develops generalized Gruss inequalities for Frechet differentiable mappings on Hilbert C*-modules, expanding the theoretical framework for inequalities in operator algebra contexts.

## Contribution

It introduces new semi-inner products on differentiable function spaces and establishes Gruss type inequalities within this generalized setting.

## Key findings

- Established Gruss inequalities in semi-inner product C*-modules.
- Introduced generalized semi-inner products for differentiable mappings.
- Extended inequalities to mappings on Hilbert C*-modules.

## Abstract

Let X be a Hilbert C^*-module on C^*-algebra A and p in A. We denote by Dp(A;X) the set of all continuous functions f on A, which are Frechet differentiable on a open neighborhood U of p. Then, we introduce some generalized semi-inner products on Dp(A;X), and using them some Gruss type inequalities in semi-inner product C^*-module Dp(A;X) and Dp(A;X^n) are established.