This note deals with observer design for Lipschitz nonlinear systems via LMI. A new LMI condition is proposed to solve the problem of H_∞ circle criterion observer design. This enhanced LMI is less conservative than those proposed in the literature for the same class of systems by using the same methodology. Indeed, thanks to a new and judicious use of the Young's relation, additional degree of freedoms are included in the LMI, contrarily to some recent results, which turn to be particular cases of what we proposed in this paper. This additional decision variables allow to improve the feasibility of the proposed LMI. A numerical example is given to show the effectiveness of the proposed methodology.