An inequality concerning the growth bound of a discrete evolution family on a complex Banach space

We prove that the uniform growth bound $\omega_0(\mathcal{U})$ of a discrete evolution family $\mathcal{U}$ of bounded linear operators acting on a complex Banach space $X$ satisfies the inequality $$\omega_0(\mathcal{U})c_{\mathcal{U}}(\mathcal{X})\le -1;$$ here $c_{\mathcal{U}}(\mathcal{X})$ is the operator norm of a convolution operator which acts on a certain Banach space $\mathcal{X}$ of $X$-valued sequences.