Roth (Bull. Aust. Math. Soc. 95, 495-499 (2017)) gave a result for DDVV inequality submanifolds of warped product manifolds. Then Murathan et al. (J. Geom. (2018) 109:30) obtained the Wintgen-like inequality for statistical submanifolds of statistical warped product manifolds. Recently, Gorunus et al. (Inter. Elec. J. Geom. 12(1), 43-56 (2019)) established the generalized Wintgen inequality for Legendrian submanifolds in almost Kenmotsu statistical manifolds. Thus, in the present manuscript, we find the generalized Wintgen inequality for a Legendrian submanifold in the standard warped product manifolds (β-Kenmotsu manifold).