论文标题
从下方的下方函数下方的限制通过整个函数模块的对数
The restriction from below of the subharmonic function by the logarithm of the module of entire function
论文作者
论文摘要
令$ u \ not \ equiv - \ infty $为复杂平面$ \ mathbb c $上的次谐波函数。然后，对于任何功能，$ r \ colon \ mathbb c \ to（0,1] $满足条件$$ \ inf_ {z {z \ in \ mathbb c} \ frac {\ ln r（z）} {\ ln r（z）} {\ ln（2+ | z | | | | |）}> - \ iftty，$ f \ $ f \ $ f \ $ f \ $ | f（z）| \ leq \ frac {1} {2π} \ int_0^{2π} u（z+r（z）表格$ \ ln | f（z）| \ leq u（z）$的不平等在\ in \ mathbb c \ setminus e $上的所有点$ z \，其中特殊的集合$ e $在$ d $ d $ d $ d $ d $ d $ d $ d $ d $ e $的$ e $中，带有可变的radius $ r $。
Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. Then for any function $r\colon\mathbb C\to (0,1]$ satisfying the condition $$\inf_{z\in\mathbb C}\frac{\ln r(z)}{\ln(2+|z|)}>-\infty,$$ there is an entire function $f\not\equiv 0$ such that $$ \ln |f(z)|\leq \frac{1}{2π}\int_0^{2π}u(z+r(z)e^{iθ})\,{\mathrm d}θ\quad\text{for all $z\in\mathbb C$.}$$ A similar result is established for subharmonic functions of finite order with inequalities of the form $\ln|f(z)|\leq u(z)$ at all points $z\in\mathbb C\setminus E$, where the exceptional set $E$ is small in terms of $d$-dimensional Hausdorff content of $E$ with variable radius $r$.
