Let $\mu(z) dV_n$ be a measure in $\mathbb{C} ^n$.
Let $B_n(r) := \{z \mid \|z\| < r\}$ be the ball of radius $r$ in $\mathbb C^n$, and $\partial B_n(r) $ be the corresponding sphere.
In $\mathbb{C} $ how can we find the following inequality?
$$
\operatorname{Vol}_{\mu}(B(r))=\int_{B_1(r)} \mu(z) dV_1(z)=
\int_0^r\left(\int_{\partial B_1(t)} \mu dz\right)dt\geq \int_0^r \left[\int_{\partial B_1(t)}(\mu)^{ \frac{1}{2}} \right]^2\frac{1}{2\pi t} dt
$$
And can we generalize this inequality in $\mathbb {C}^n$?