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Growth of solutions of second order linear differential equations. (English) Zbl 0952.34070

Consider the linear differential equation

f '' +A(z)f ' +B(z)f=0,(1)

where A(z),B(z)¬0 are entire functions satisfying ρ(B)<ρ(A)· The following question is natural: If A(z) has no finite deficient values, does every nonconstant solution to (1) have infinite order? The authors study the growth of solutions to (1) under a condition related to this question. The main result is stated as follows: If ρ(B)<ρ(A)< and

T(r,A)/logM(r,A)1(2)

as r outside a set of finite logarithmic measure, then every nonconstant solution to (1) has infinite order. By T. Murai [Ann. Inst. Fourier 33, No. 3, 39-58 (1983; Zbl 0519.30029)], if A(z) has Fejér gaps, then (2) is valid for some exceptional set of finite logarithmic measure. Hence the same conclusion holds, if ρ(B)<ρ(A)< and A(z) has Fejér gaps.


MSC:
34M10Oscillation, growth of solutions (ODE in the complex domain)
34M05Entire and meromorphic solutions (ODE)
30D20General theory of entire functions
30D35Distribution of values (one complex variable); Nevanlinna theory
30D05Functional equations in the complex domain, iteration and composition of analytic functions
34M20Nonanalytic aspects differential equations in the complex domain (MSC2000)