A Variational link between the Olech-Opial inequality, the Wirtinger inequality, and Emden-Fowler equations

We establish a structural connection between the classical Olech-Opial inequality and the Wirtinger inequality. Using an integral identity involving the mixed energy term $u{u}^{\mathrm{\prime }}$, we derive a nonlinear interpolation inequality linking these two results. The optimal constant is characterized by a variational problem whose extremals satisfy an Emden-Fowler equation. An explicit expression of the optimal constant is obtained in terms of the Beta function. This approach provides a natural bridge between mixed-energy integral inequalities, classical spectral estimates, and nonlinear boundary value problems.