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On subnormal solutions of second order linear periodic differential equations. (English) Zbl 1132.34064

The authors study the growth of solutions of the linear periodic differential equation

f '' +(P 1 (e z )+P 2 (e -z ))f ' +(Q 1 (e z )+Q 2 (e -z ))f=R 1 (e z )+R 2 (e -z )(1)

and its corresponding homogeneous equation

f '' +(P 1 (e z )+P 2 (e -z ))f ' +(Q 1 (e z )+Q 2 (e -z ))f=0,(2)

where P j (z),Q j (z),R j (z) (j=1,2) are polynomials in z. It is well-known that all solution of (2) are entire functions. Let σ(f) denote the order of growth of f(z) and use the notation σ 2 (f) to denote the hyper-order of f(z).

Theorem 1. If

degQ 1 >degP 1 ordegQ 2 >degP 2 ,(3)

then the differential equation (2) has no nontrivial subnormal solution, and every solution of (2) satisfies σ 2 (f)=1.

Theorem 2. If R 1 +R 2 0 and P j ,Q j (j=1,2) satisfy (3), then (i) (1) has at most one subnormal solution f 0 , and f 0 =S 1 (e z )+S 2 (e -z ), where S 1 (z) and S 2 (z) are polynomials in z; (ii) all other solutions f of (1) satisfy σ 2 (f)=1 except a subnormal solution in (i).

Theorem 2 refines and generalizes a result due to G. Gundersen and M. Steinbart for P 2 =Q 2 =0.


MSC:
34M10Oscillation, growth of solutions (ODE in the complex domain)
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