Direct analogues of Wiman's inequality for analytic functions in the unit disc
Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be an analytic function on $\{z:|z|<1\},\ h\in H$ and $\Omega_f(r)= \sum_{n=0}^{\infty} |a_n| r^n$. If<br />$$<br />\beta_{fh}=\liminf\limits_{r\to1}\frac{\ln\ln\Omega_f(r)}{\ln h(r)}=+\infty,<br />$$<br />then Wiman's inequa...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Vasyl Stefanyk Precarpathian National University
2013-01-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Online Access: | http://journals.pu.if.ua/index.php/cmp/article/view/51 |
| Summary: | Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be an analytic function on $\{z:|z|<1\},\ h\in H$ and $\Omega_f(r)= \sum_{n=0}^{\infty} |a_n| r^n$. If<br />$$<br />\beta_{fh}=\liminf\limits_{r\to1}\frac{\ln\ln\Omega_f(r)}{\ln h(r)}=+\infty,<br />$$<br />then Wiman's inequality $M_f(r)\leq \mu_f(r) \ln^{1/2+\delta}\mu_f(r)$ is true for all $r\in (r_0, 1)\backslash E(\delta)$, where $h-\mbox{meas}\ E<+\infty.$ |
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| ISSN: | 2075-9827 2313-0210 |