Some Grüss Type Inequalities for Fréchet Differentiable Mappings

T. Teimouri-Azadbakht, A. G Ghazanfari

Let $X$ be a Hilbert $C^{*}$-module on $C^{*}$-algebra $A$ and $p\in A$. We denote by $D_{p}(A ,X)$ the set of all continuous functions $f : A \rightarrow X$, which are Fréchet differentiable on a open neighborhood $U$ of $p$. Then, we introduce some generalized semi-inner products on $D_{p}(A ,X)$, and using them some Grüss type inequalities in semi-inner product $C^*$-module $D_{p}(A ,X)$ and $D_{p}(A ,X^n)$ are established.