Bautin’s approach to the bifurcation theory of limit cycles has been recently generalized in the framework of complex analysis. There are now more cases where the Bautin ideal is known. A systematic study of the Poincar ́e center focus problem via Abel equations entailed several new examples of Bautin ideals This article deals with Li ́enard equations which have been used in many applications (c.f. [10]). Li ́enard equations play certainly a key role in Hilbert’16th problem, because of the topological simplicity of the return mapping. Limit cycles encircle the origin and are necessarily contained in the domain of existence of the return mapping. This domain of existence may of course not be equal to the domain of convergence of the analytic series which defines the return mapping in a neighborhood of the origin. Nevertheless it is interesting to produce an estimate of the size of this domain of convergence.