This article addresses the \(H_\infty \) state estimation problem of generalized neural networks (GNNs) subject to mixed delays. Firstly, an extended integral inequality is newly proposed by combining the third-order generalized free-matrix-based inequality (GFMBI) and an improved reciprocally convex lemma (IRCL) into a unified frame. Secondly, to coordinate with the features of the new developed integral inequality, a modified Lyapunov–Krasovskii functional (LKF) with the consideration of more information on mixed delays and nonlinear activation function is established. Thirdly, by applying the proposed integral inequality and finite-interval quadratic polynomial inequality, a further enhanced state estimation criterion is achieved to design suitable \(H_\infty \) state estimator gains. Finally, two well-studied simulations examples are done to illustrate the validity of the proposed approach.
