Summary: In this paper, we introduce a new class of warped products, called generic warped product submanifolds in locally product Riemannian manifolds with pointwise slant fiber. We prove that every generic warped product submanifold \(B \times_f M_\theta\) in a locally product Riemannian manifold satisfies the following inequality:
\[\|h\|^2 \geq s\bigl[\cos^2 \theta \|\overset{\rightarrow}{\nabla}^\bot(\ln f)\|^2 + 2 (\csc \theta + \cot \theta)^2 \|\overset{\rightarrow}{\nabla}^T(\ln f)\|^2\bigr]\]
where \(B = M_T \times M_\bot\), a semi-invariant submanifold; \(M_\theta\) is a pointwise slant submanifold of dimension \(s\) and \(\overset{\rightarrow}{\nabla}^T(\ln f)\) and \(\overset{\rightarrow}{\nabla}^\bot(\ln f)\) are gradient components of the warping function \(\ln f\) along \(M_T\) and \(M_\bot\), respectively. The equality case of the lower bound is also considered. Furthermore, we give many applications of this inequality and construct some non-trivial examples.