Spectrally bounded $\phi$-derivations on Banach algebras

Applying the density theorem on algebras with $\phi$-derivations, we show that if a $\phi$-derivation $\delta$ of a unital Banach algebra $A$ is spectrally bounded, then $[\delta (A), A]\subseteq \text{rad}(A)$. Also, $\delta (A)\subseteq \text{rad}(A)$ if and only if $\text{sup}\{r(z^{-1}\delta (z))\mid z\in A \text{is invertible}\}<\infty$, where $r(a)$ denotes the spectral radius of $a\in A$.