线性微分方程解的亏值
Deficient values of solutions of linear differential equations
摘要：
考虑形式为$f'' + A(z)f' + B(z)f = 0$（*）的微分方程，其中$A(z)$和$B(z) \not\equiv 0$是整函数.用Lindelöf函数证明了对于任何$ρ\in (1/2, \infty)$，存在一个形式为（*）的方程，它有一个$ρ$阶的解$f$，在$0$处有一个Nevanlinna亏值，其中$f, A(z), B(z)$满足一个共同的增长条件.已知当$ρ\leq 1/2$。对于较小的增长函数，Anderson和Clunie的一个例子的几何修改被用来表明，对于任何$ρ\in (2, \infty)$，存在一个形式为（*）的方程，该方程具有对数阶$ρ$的解$f$，其Valiron亏值为$0$，其中$f, A(z), B(z)$满足类似的增长条件。这个结果基本上是尖锐的。在这两个证明中，所示解的零点的分离起着关键作用。对线性微分方程解的亏值也给出了一些观察，其中包括对Nevanlinna亏值的Wittich定理的讨论，对Valiron亏值的一个修正的Wittich定理，Gol'dberg定理的后果，并举例说明可能发生的可能性。
英文摘要：
Differential equations of the form $f'' + A(z)f' + B(z)f = 0$ (*) are considered, where $A(z)$ and $B(z) \not\equiv 0$ are entire functions. The Lindelöf function is used to show that for any $ρ\in (1/2, \infty)$, there exists an equation of the form (*) which possesses a solution $f$ of order $ρ$ with a Nevanlinna deficient value at $0$, where $f, A(z), B(z)$ satisfy a common growth condition. It is known that such an example cannot exist when $ρ\leq 1/2$. For smaller growth functions, a geometrical modification of an example of Anderson and Clunie is used to show that for any $ρ\in (2, \infty)$, there exists an equation of the form (*) which possesses a solution $f$ of logarithmic order $ρ$ with a Valiron deficient value of at $0$, where $f, A(z), B(z)$ satisfy an analogous growth condition. This result is essentially sharp. In both proofs, the separation of the zeros of the indicated solution plays a key role. Observations on the deficient values of solutions of linear differential equations are also given, which include a discussion of Wittich's theorem on Nevanlinna deficient values, a modified Wittich theorem for Valiron deficient values, consequences of Gol'dberg's theorem, and examples to illustrate possibilities that can occur.
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