论文标题
复杂LDE的单位光盘中复杂LDE的不规则有限级解决方案
Irregular finite order solutions of complex LDE's in unit disc
论文作者
论文摘要
结果表明，对于$ f^{（k）}+ a f = 0 $的所有非客气解决方案的顺序和较低的增长率是相等的，并且仅当设备光盘中的系数$ a $是分析性的，$ \ log^+ m（r，a）/\ log（a）/\ log（1-r）$倾向于$ r r \ r \ t o $ r \ tos $ r \ tos $ r \ to $ r \。构建了一个混凝土示例家族，其中解决方案的顺序保持不变，而较低顺序可能会根据系数的不规则生长而在一定间隔内变化。这些系数随着模量的对数近似于在足够大的单位盘子集上的规定不规则生长的平滑径向亚谐波功能。还建立了描述这些高度不平凡的例子背后现象的结果。在达到一般性结果的途径中，发现了一种新的尖锐的对数衍生物估计，涉及较低的生长。除了这些估计之外，所使用的参数尤其是基于适合低阶的Wiman-Valiron理论，以及对最大模量对数的右衍生的充分理解。
It is shown that the order and the lower order of growth are equal for all non-trivial solutions of $f^{(k)}+A f=0$ if and only if the coefficient $A$ is analytic in the unit disc and $\log^+ M(r,A)/\log(1-r)$ tends to a finite limit as $r\to 1^-$. A family of concrete examples is constructed, where the order of solutions remain the same while the lower order may vary on a certain interval depending on the irregular growth of the coefficient. These coefficients emerge as the logarithm of their modulus approximates smooth radial subharmonic functions of prescribed irregular growth on a sufficiently large subset of the unit disc. A result describing the phenomenon behind these highly non-trivial examples is also established. En route to results of general nature, a new sharp logarithmic derivative estimate involving the lower order of growth is discovered. In addition to these estimates, arguments used are based, in particular, on the Wiman-Valiron theory adapted for the lower order, and on a good understanding of the right-derivative of the logarithm of the maximum modulus.
