How conservative is the circle criterion?

Abstract

The following notation and terminology will be used. By a stable transfer function we mean a rational scalar transfer function G = G(z) with real coefficients and no poles in the area |z| ≥ 1. ‖G‖_∞ denotes the so-called H-infinity norm of G, defined as

$${\left\| G \right\|_\infty } = \mathop {\max }\limits_{\left| z \right| = 1} \left| {G(z)} \right|.$$