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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 $\mathbb {C}^n$?

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? $$ \operatorname{Vol}_{\mu}(B_1(r))=\int_{B_1(r)} \mu(z) dV_1(z)= \int_0^r\left(\int_{\partial B_1(t)} \mu dz\right)dt\geq \int_0^r \left[\int_{\partial B_1(t)}(\mu)^{ \frac{1}{2}} \right]^2\frac{1}{2\pi t} dt $$ And can we generalize this inequality in $\mathbb {C}^n$?

Let $\mu(z) dV$ be a measure in $\mathbb{C} ^n$. Let $B_{\mu}(r) $ be a ball in $\mathbb{C} ^n$, and $\partial B_{\mu}(r) $ be the sphere. In $\mathbb{C} $ how can we find the following inequality? $$ \operatorname{Vol}(B_{\mu}(r))=\int_{B_{\mu}(r)} \mu(z) dV= \int_0^r\int_{\partial B_{\mu}(t)} \mu dt\geq \int_0^r \left[\int_{\partial B_{\mu}(t)}(\mu)^{ \frac{1}{2}} \right]^2\frac{1}{2\pi t} dt $$ And can we generalize this inequality in $\mathbb {C} ^n$?

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 $\mathbb {C}^n$?

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Let $\mu(z) dV$ be a measure in $\mathbb{C} ^n$. Let $B_{\mu}(r) $ be a ball in $\mathbb{C} ^n$, and $\partial B_{\mu}(r) $ be the sphere. In $\mathbb{C} $ how can we find the following inequality? $$ \operatorname{Vol}(B_{\mu}(r))=\int_{B_{\mu}(r)} \mu(z) dV= \int_0^r\int_{\partial B_{\mu}(t)} \mu dt\geq \int_0^r \left[\int_{\partial B_{\mu}(t)}(\mu)^{ \frac{1}{2}} \right]^2\frac{1}{2\pi t} dt $$ And can we generalize this inequality in $\mathbb {C} ^n$?