Refined nonlinear Caputo-Hadamard fractional Halanay inequalities with sharp decay rates

We establish new Halanay-type inequalities for systems involving the Caputo-Hadamard fractional derivative with unbounded time-varying delays, extending the results of [J. Math. Anal. Appl., 525(1):127145, 2023]. For the linear case, we derive a sharp and explicit decay rate under delay condition (D), thereby resolving the open problem posed in [Fract. Calc. Appl. Anal. 25(6):2420-2445, 2022]. For homogeneous power-type nonlinearities, we obtain refined upper bounds and, in the superlinear regime, also establish matching lower bounds, demonstrating that the decay explicitly depends on both the fractional order and the nonlinearity exponent. As an application, we prove Mittag-Leffler stability for a class of Hopfield-type neural networks with proportional delays. These results extend and sharpen existing inequalities in the literature and provide effective tools for the stability analysis of fractional-order systems with delays.