AI 中文总结
该研究针对单位圆盘内满足非齐次解析伸缩方程的保向调和映射，在相应约束条件下建立了Bloch型定理，并在附加假设下得到两种版本的Landau定理，丰富了调和映射相关理论。
AI 中文摘要
我们研究单位圆盘$\u27d9$中一类保向调和映射$f=h+\u0305{g}$的Bloch型与Landau型定理，这类映射满足非齐次解析伸缩方程$g'(z)=ω(z)h'(z)+ψ(z)$，其中$ω$和$ψ$是$\u27d9$内的解析函数，满足$\u2016ω\u2016_{\u221e}\u2a7d k<1$且$\u2016ψ\u2016_{\u221e}\u2a7d M$。这里$h$和$g$分别称为$f$的解析部分与余解析部分。我们首先在$k+M<1$的条件下，为某类规范化调和函数建立了Bloch型定理。最后，我们在附加假设下得到了两种版本的Landau定理：一种针对有界调和映射，另一种则基于调和函数的解析部分具有有界Bloch半范数的假设。
英文摘要
We study Bloch and Landau type theorems for a class of sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying the inhomogeneous analytic dilatation equation $$ g'(z)=ω(z)h'(z)+ψ(z),$$ where $ω$ and $ψ$ are analytic functions in $\mathbb{D}$ with $\|ω\|_{\infty}\leq k<1$ and $\|ψ\|_{\infty}\leq M$. Here $h$ and $g$ are called analytic and co-analytic part of $f$, respectively. We first establish a Bloch type theorem for certain normalized class of harmonic functions under the condition $k+M<1$. Finally, we obtain two versions of the Landau theorem under additional assumptions: one for bounded harmonic mappings and another under the assumption that the analytic part of a harmonic functions has bounded Bloch seminorm.
Comments10 pages
