The answer is no. Consider $B\subset A$ with $\mu(B)$ very small, $C_1=A\setminus B$ and  $C_2=X\setminus C_1$. Then the r.h.s. equals $\mu(B)^2/\mu(A)\mu(C_2)$ which can be less than $\mu(B)$ since $\mu(A)$ and $\mu(C_2)$ can be grater than $1/2$ and $\mu(B)$ less than $1/10$.

I wonder what makes you think that this strange inequality could possibly be true...