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Bull. Korean Math. Soc. 2024; 61(4): 959-968

Online first article May 16, 2024      Printed July 31, 2024

https://doi.org/10.4134/BKMS.b230535

Copyright © The Korean Mathematical Society.

The growth of Bloch functions in some spaces

Wenwan Yang, Junming Zhugeliu

Guangdong University of Technology; Guangdong University of Technology

Abstract

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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 \mathrm{e}^{\mathrm{i} t})|^p\mathrm{d} t\right)^{1/p} \\
&\le \left(\frac{\Gamma(\frac p2+1)}{\Gamma(\frac p2+1-k)} \right)^{1/p} \rho_\mathcal{B} \left(\log\frac1{1-r^2}\right)^{1/2},
\end{align}
where $\rho_\mathcal{B}(f)=\sup_{z\in\mathbb{D}}(1-|z|^2)|f'(z)|$ and $k$ is the   integer satisfying $0
  Moreover, we prove that for $01$,
$$\|f_r\|_{B_q}\le r\, \rho_\mathcal{B}(f) \left(\frac{1}{(1-r^2)(q-1)}\right)^{1/q},$$
where
$f_r(z)=f(rz)$ and  $\|\cdot\|_{B_q}$ is the  Besov seminorm given by
$$\|f\|_{B_q}=\left(\int_{\mathbb{D}}|f'(z)|^q(1-|z|^2)^{q-2}\mathrm{d} A(z)\right).$$
 These results improve previous results of   Clunie and   MacGregor.

Keywords: Bloch spaces, integral mean, Besov space

MSC numbers: Primary 30H30

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