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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 {\it 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.

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)
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