This paper introduces a Wirtinger-based integral inequality as an alternative to the Jensen inequality, which is commonly used in the stability analysis of time-delay and sampled-data systems. The authors propose a new inequality that encompasses the Jensen inequality and leads to tractable linear matrix inequality (LMI) conditions. This new inequality is derived using Fourier theory and Wirtinger inequalities, which are known for their accuracy in estimating integral quadratic terms. The paper demonstrates the potential of this new inequality by providing stability criteria for time-delay and sampled-data systems, showing that it reduces conservatism compared to the Jensen inequality. Two applications are presented to illustrate the effectiveness of the proposed inequality, including stability analysis for systems with constant and known delays, and systems with time-varying delays. The results are compared with existing methods, demonstrating the competitiveness and accuracy of the new inequality.This paper introduces a Wirtinger-based integral inequality as an alternative to the Jensen inequality, which is commonly used in the stability analysis of time-delay and sampled-data systems. The authors propose a new inequality that encompasses the Jensen inequality and leads to tractable linear matrix inequality (LMI) conditions. This new inequality is derived using Fourier theory and Wirtinger inequalities, which are known for their accuracy in estimating integral quadratic terms. The paper demonstrates the potential of this new inequality by providing stability criteria for time-delay and sampled-data systems, showing that it reduces conservatism compared to the Jensen inequality. Two applications are presented to illustrate the effectiveness of the proposed inequality, including stability analysis for systems with constant and known delays, and systems with time-varying delays. The results are compared with existing methods, demonstrating the competitiveness and accuracy of the new inequality.