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Growth of solutions of second order linear differential equations. (English) Zbl 0952.34070
Consider the linear differential equation \[ f''+A(z)f'+B(z)f=0, \tag{1} \] where \(A(z), B(z)\not\equiv 0\) are entire functions satisfying \(\rho(B)<\rho(A).\) 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 \(\rho(B)< \rho(A)<\infty\) and \[ T(r,A)/\log M(r,A)\to 1 \tag{2} \] as \(r\to\infty\) 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 \(\rho(B)<\rho(A)<\infty\) and \(A(z)\) has Fejér gaps.

34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
30D20 Entire functions of one complex variable, general theory
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
34M20 Nonanalytic aspects differential equations in the complex domain (MSC2000)
Full Text: DOI
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