This paper deals with the nonlinear H-infinity-control problem with sampled measurement feedback. This problem has already been studied in [10], where, using a certainty equivalence principle, a control solution involving a state estimator with a linear injection gain is proposed. Using the same general framework, we propose a more refined estimator with a nonlinear injection gain. This gain is shown to be connected to a periodic solution of a Hamilton-Jacobi inequality with jumps.