  We estimate the norm of the almost Mathieu operator $H_{\theta,\lambda}
=U+U^*+(\lambda /2)(V+V^*)$ in the rotation $C^*$-algebra $A_\theta =C^*(U,V
unitaries;UV=e^{2\pi i\theta} VU)$. In this process, we significantly improve
the inequality $\| H_\theta \| \leq 2\sqrt{2}$, $\theta \in [0.25,0.5]$,
conjectured by Beguin, Valette and Zuk.
