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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