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On the complex oscillation of some linear differential equations. (English) Zbl 0877.34009

The paper treats the linear differential equation,

f (k) +A(z)f=0,(*)

where k2 is an integer, A(z) is a transcendental entire function of order σ(A). It is shown that any non-trivial solution of (*) satisfies λ(f)σ(A), where λ(f) is the exponent of convergence of the zero-sequence of f, under the condition,

KN ¯r , 1 AT(r,A),rE

for K>2k and an exceptional set, E, of finite linear measure. Herein N(r,f) and T(r,f) denotes the counting function and the characteristic function of f respectively. A very nice example is given demonstrating the result. Several technical lemmas are extremely well done preparing the proof of the theorem.

The other half of the paper treats the second-order equation,

f '' +(e p 1 (z) +e p 2 (z) +Q(z))f=0,(**)

where p 1 (z) and p 2 (z) are non-constant polynomials of degree n and m respectively. Q(z) is an entire function of order less than max(m,n). Several theorems are proved regarding equation (**), where once again several well done lemmas prepare the proof of the theorem. The paper is very well written.


MSC:
34M10Oscillation, growth of solutions (ODE in the complex domain)
30D05Functional equations in the complex domain, iteration and composition of analytic functions