The growth of Bloch functions in some spaces
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Suppose $f$ belongs to the Bloch space with $f(0)=0$. For $0<r<1$ and $0<p<\infty$, we show that \begin{align*} M_p(r,f)&=\left(\frac1{2\pi}\int_0^{2\pi}|f(r \rme^{\rmi t})|^p\rd t\right)^{1/p} \\ &\le \left(\frac{\Gamma(\frac p2+1)}{\Gamma(\frac p2+1-k)} \right)^{1/p} \rho_\cb \left(\log\frac1{1-r^2}\right)^{1/2}, \end{align*} where $\rho_\cb(f)=\sup_{z\in\bd}(1-|z|^2)|f'(z)|$ and $k$ is the integer satisfying $0<p-2k\le2$. Moreover, we prove that for $0<r<1$ and $p>1$, $$\|f_r\|_{B_q}\le r\, \rho_\cb(f) \left(\frac{1}{(1-r^2)(q-1)}\right)^{1/q},$$ where $f_r(z)=f(rz)$ and $\|\cdot\|_{B_q}$ is the Besov seminorm given by $$\|f\|_{B_q}=\left(\ibd|f'(z)|^q(1-|z|^2)^{q-2}\rd A(z)\right).$$ These results improve previous results of Clunie and MacGregor.
Abstract
Suppose $f$ belongs to the Bloch space with $f(0)=0$. For $0<r<1$ and $0<p<\infty$, we show that \begin{align*} M_p(r,f)&=\left(\frac1{2\pi}\int_0^{2\pi}|f(r \rme^{\rmi t})|^p\rd t\right)^{1/p} \\ &\le \left(\frac{\Gamma(\frac p2+1)}{\Gamma(\frac p2+1-k)} \right)^{1/p} \rho_\cb \left(\log\frac1{1-r^2}\right)^{1/2}, \end{align*} where $\rho_\cb(f)=\sup_{z\in\bd}(1-|z|^2)|f'(z)|$ and $k$ is the integer satisfying $0<p-2k\le2$. Moreover, we prove that for $0<r<1$ and $p>1$, $$\|f_r\|_{B_q}\le r\, \rho_\cb(f) \left(\frac{1}{(1-r^2)(q-1)}\right)^{1/q},$$ where $f_r(z)=f(rz)$ and $\|\cdot\|_{B_q}$ is the Besov seminorm given by $$\|f\|_{B_q}=\left(\ibd|f'(z)|^q(1-|z|^2)^{q-2}\rd A(z)\right).$$ These results improve previous results of Clunie and MacGregor.
참고문헌 (16)
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