How can I prove the following inequality: (where $f $ is nice enough) -
Given a function $ f(x,y) : \Omega_1 \times \Omega_2 \to \mathbb{R} $ , and $\alpha,C_1,C_2 $ are some constants, ( $\Omega_i$ is equipped with a probability measure $ \mu_i $ respectively) , then there exists a constant $C_3 $ such that $ \begin{multline}C_1^2 \int_{\Omega_1 } \left| \int_{\Omega_2} f(x,y) d \mu _2 - \alpha\right| ^2 d\mu_1 + C_2^2 \int_{\Omega_2 } \left| \int_{\Omega_1} f(x,y) d \mu _1 - \alpha\right|^2 d\mu_2 \\ \geq C_3^2 \int_{\Omega_1}\int_{\Omega_2} |f(x,y)-\alpha|^2 d\mu_1 d\mu_2\end{multline}$ is true.
It seems like it's kind of triangle's inequality. Have you got an idea?