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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.

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