Let A be a complex unital Banach algebra with unit 1. An element $$a\in A$$ is said to of $$G_{1}$$ -class if $$\begin{aligned} \Vert (z-a)^{-1}\Vert =\frac{1}{\text {d}(z,\sigma (a))} \quad \forall z\in {\mathbb {C}}\setminus \sigma (a). \end{aligned}$$ Here $$d(z, (a))$$ denotes the distance between z and spectrum $$\sigma (a)$$ a. Some examples such elements are given also some properties pr...