This paper is devoted to fuzzy dynamic output feedback control for nonlinear networked multirate sampled-data systems based on T-S fuzzy systems. Using asynchronous multirate sampled-data, a multirate fuzzy controller is designed. The resultant closed-loop system is modeled as a perturbed system with two perturbed terms which include multirate sampling and the asynchronous membership functions between the controlled plant and the controller. A novel refined integral inequality is developed, and it improves Jensen's inequality and extends Wirtinger's inequalities by considering the sawtooth structure of time-varying delays. By applying the inequality to the perturbed system and constructing a new storage function, a stability criterion and two fuzzy dynamic output feedback controller design methods are presented for the nonlinear networked multirate sampled-data systems. Finally, three illustrative examples show the effectiveness of the proposed method.