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首页> 外文期刊>Canadian Journal of Mathematics >Normal Functions: $L^p$ Estimates
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Normal Functions: $L^p$ Estimates

机译:正常函数:$ L ^ p $估计

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For a meromorphic (or harmonic) function $f$, let us call the dilation of $f$ at $z$ the ratio of the (spherical) metric at $f(z)$ and the(hyperbolic) metric at $z$. Inequalities are known which estimatethe $sup$ norm of the dilation in terms of its $L^p$ norm, for $p>2$,while capitalizing on the symmetries of $f$. In the present paperwe weaken the hypothesis by showing that such estimates persisteven if the $L^p$ norms are taken only over the set of $z$ on which$f$ takes values in a fixed spherical disk. Naturally, the biggerthe disk, the better the estimate. Also, We give estimates forholomorphic functions without zeros and for harmonic functions inthe case that $p=2$.
机译:对于亚纯函数(或谐波)$ f $,我们称$ f $在$ z $处的膨胀是(f)球形度量在$ f(z)$与(双曲)度量在$ z $处的比率。已知不等式,它利用$ f $的对称性,根据$ L ^ p $范数来估计膨胀的$ sup $范数(对于$ p> 2 $)。在本文中,我们通过证明即使$ L ^ p $范数仅在固定球盘上$ f $取值的$ z $集合上采用,这种估计仍然存在,从而削弱了这一假设。自然,磁盘越大,估计越好。此外,在$ p = 2 $的情况下,我们给出了不带零的全纯函数的估计以及谐波函数的估计。

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