On the oscillation of solutions of certain linear differential equations in the complex domain. (English) Zbl 0596.30049

Bank, Steven B.; Langley, J. K.

Proc. Edinb. Math. Soc., II. Ser. 30, 455-469 (1987).

Let A(z) be an entire function of finite order having zero as a Borel exceptional value. Conditions are given on the location of zeros of A(z) which ensure that all non-trivial solutions of \[ y^{(k)}+(A(z)+Q(z))y=0, \] where \(k\geq 2\) and Q is a sufficiently small polynomial, have zeros with infinite exponent of convergence. In particular this is true for all solutions of \(y^{(k)}+e^{P(z)}y=0,\) if \(k\geq 2\) and P is a non-constant polynomial.

MSC:

30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M99	Ordinary differential equations in the complex domain

Keywords:

Borel exceptional value

References:

[1]	Bank, Resultate Math.
[2]	DOI: 10.2307/1999210 · Zbl 0505.34026 · doi:10.2307/1999210
[3]	Bank, Math. Scand. 40 pp 119– (1977) · Zbl 0367.34004 · doi:10.7146/math.scand.a-11681
[4]	DOI: 10.1007/BF01176476 · Zbl 0494.34005 · doi:10.1007/BF01176476
[5]	Bellman, Stability Theory of Differential Equations (1953)
[6]	DOI: 10.1007/BF01580359 · Zbl 0226.34005 · doi:10.1007/BF01580359
[7]	Hayman, Meromorphic Functions (1964)
[8]	DOI: 10.1007/BF02565929 · Zbl 0123.26702 · doi:10.1007/BF02565929
[9]	Titchmarsh, The Theory of Functions (1949) · Zbl 0035.18001