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Growth of solutions of higher order linear differential equations. (English) Zbl 1306.34135
Let \(m(r,f)\) be the proximity function of a meromorphic function \(f\) in the complex plane, \(N(r,f)\) be its Nevanlinna’s counting function. Let \(\rho(f)\) denote the order of the Nevanlinna characteristic \(T(r,f)=m(r,f)+N(r,f)\). For a meromorphic function \(\varphi\), let \[ \lambda(f-\varphi)=\limsup_{r\to+\infty} \frac{\log N(r, 1/(f-\varphi)}{\log r}, \lambda_2(f-\varphi)=\limsup_{r\to+\infty} \frac{\log \log N(r, 1/(f-\varphi)}{\log r}, \] \(\bar \lambda\) and \(\bar \lambda_2\) be the corresponding values defined by the reduced function \(\bar N\), respectively.
The main result of the paper is the following theorem.
Theorem 1.5. Let \(P(z)=\sum_{j=0}^n a_j z^j\) and \(Q(z)=\sum_{j=0}^n b_j z^j\) be nonconstant polynomials, where \(a_j, b_j\) (\(j=0, 1,\dots, n\)) are complex numbers, \(a_nb_n (a_n-b_n)\neq 0\). Suppose that \(A_j(z)\) (\(j=0, 1,\dots, k-1\)), \(A_j(z)\not\equiv 0\), \((j=0,1)\), \(D_j(z)\) and \(F(z)\) are entire functions with \[ \max\{ \max_j \{\rho(A_j), \rho(D_j)\}, \rho (F)\}<n \] and let \(\varphi(z)\not\equiv 0\) be an entire function of finite order. Then every solution \(f\not\equiv 0\) of the equation \[ f^{(k)}+A_{k-1}f^{(k-1)} + \dots +A_2f''+ (D_1(z)+A_1(z)e^{P(z)})f' +(D_0(z)+A_0(z)e^{Q(z)})f=F, \quad k\geq 2, \eqno(15) \] satisfies \[ \bar \lambda(f-\varphi)=\rho(f)=\infty, \quad \bar \lambda _2(f-\varphi)=\rho_2(f)\leq n. \] Furthermore if \(F\not\equiv 0\), then every solution \(f\) of equation (15) satisfies \[ \lambda (f)=\bar \lambda(f)=\rho(f)=\infty \] and \[ \lambda _2(f)=\bar \lambda _2(f) =\bar \lambda _2(f-\varphi)=\rho_2(f)\leq n. \]
The authors also consider the case when \(\rho(F)\geq n\).
It generalizes a result due to H.-Y. Xu and T.-B. Cao [Electron. J. Qual. Theory Differ. Equ. 2009, Paper No. 63, 18 p. (2009; Zbl 1198.34189)].

MSC:
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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