Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Online first article May 16, 2024

Copyright © The Korean Mathematical Society.

The growth of Bloch functions in some spaces

Wenwan Yang and Junming Zhugeliu

Guangdong University of techlogy

Abstract

Suppose $f$ belongs to the Bloch space with $f(0)=0$. For $0 \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 Moreover, we prove that for $01$,
$$\|f_r\|_{B_q}\le r\, \rho_\cb(f) \left(\frac{1}{(1-r^2)(q-1)}\right)^{1/q}$$
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.

Keywords: Bloch spaces, Integral mean, Besov space

MSC numbers: 30H30

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