Liouville-type theorems outside of small exceptional sets for functions of finite order

Liouville-type theorems with constraints outside of small sets on circles or spheres for functions of finite order

We prove that subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some asymptotically small sets on spheres, are bounded from above everywhere. It follows that subharmonic functions of finite order on the complex plane, entire and plurisubharmonic functions of finite order, and convex or harmonic functions of finite order bounded from above outside of such sets on spheres are constant. Our results and methods of proof are also new for functions of one complex variable.

To the Liouville theorem for entire functions of finite order

We present a new, short and independent proof of the Liouville-type theorem for entire and subharmonic functions of finite order bounded outside some set of zero planar density.

The Logarithm of the Modulus of an Entire Function as a Minorant for a Subharmonic Function outside a Small Exceptional Set

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. In 2016, we obtained a result on the existence of an entire function $f\neq 0$ satisfying the estimate $\log|f|\leq {\sf B}_u$ on $\mathbb C$, where functions ${\sf B}_u$ are integral averages of $u$ for rapidly shrinking disks as it approaches infinity. We give another equivalent version of this result with $\log |f|\leq u$ outside a very small exceptional set if $u$ is of finite order.

间隙幂级数的最小模与例外集的h-测度

The minimum modulus of gap power series and h-measure of exceptional sets

对于形式为$F(z)=\sum\limits_{n=0}^{+\infty} a_{n}e^{zλ_n},\ 0\leλ_n\uparrow+\infty\ (n\to+\infty)$的整个Dirichlet级数，我们建立了关系$$ F(x+iy)=(1+o(1))a_{ν(x,F)}e^{(x+iy)λ_{ν(x,F)}} $$为真的条件作为$x\to+\infty$在某个集合$E$之外，使得$\text{ h-meas }(E)=\int_{E}dh(x)<+\infty$在$y\in\Bbb{R}$中一致，其中$h(x)$是在$[0,+\infty)$上递增到$+\infty$且不递减到$+\infty$导数的正连续函数.

Chebyshev度量中解析函数模的乘积逼近下调和函数的精度

Accuracy of approximation of subharmonic functions by logarithms of moduli of analytic ones in Chebyshev metrics

已知有穷级$ρ$的下调和函数可以通过在直到$C\log|z|$的例外集之外的点$z$处的整函数的模的对数来近似。在这篇文章中，我们证明，如果这样的近似是更精确的，i。e.常数$C$减小，然后，从$C=ρ/4$开始，异常集的大小显著增大。对无穷级的下调和函数和单位圆内的次调和函数也证明了类似的结果。这些定理改进和补充了Yulmukhametov的一个结果。

广义度量空间中准共形映射的例外集 II

Exceptional Sets for Quasiconformal Mappings in General Metric Spaces II

研究广义度量空间中准共形映射的例外集，证明在满足p-Poincaré不等式条件下，允许例外集的codimension为p，并证明该结论的最优性。