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On the hyperorder of solutions of higher order differential equations. (English) Zbl 1047.30019
This paper is devoted to proving that under certain very special cases, all transcendental solutions $$f$$ of the differential equation $f^{(k)}+H_{k-1}(z)f^{(k-1)}+\dots+H_1(z)f'+H_0(z)f=0$ are of infinite order of growth. In the main result, the assumptions are as follows: Each $$H_j(z)=h_j(z)e^{\alpha_jz}$$, where $$h_j$$ is a polynomial and $$\alpha_j\in\mathbb{C}$$. Moreover, at least for some $$s<l$$, $$h_s$$ and $$h_l$$ are nonvanishing, and $$\alpha_s=d_se^{i\varphi}$$, $$\alpha_l=-d_le^{i\varphi}$$, where $$d_s>0$$, $$d_l>0$$. In addition, for $$j\neq s$$, either $$\alpha_j=d_je^{i\varphi}$$, or $$\alpha_j=-d_je^{i\varphi}$$ for $$d_j\geq0$$, and $$\max\{\,d_j\mid j\neq s,l\,\}=:d<\min\{d_s,d_l\}$$. More precisely, it will be proved that under these conditions, the iterated order $$\rho_2(f):=\limsup_{r\to\infty}\frac{\log\log T(r,f)}{\log r}=1$$. The proof makes use of basic results from the Wiman–Valiron theory as well as standard estimates for generalized logarithmic derivatives.

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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##### References:
  doi:10.1515/9783110863147 · doi:10.1515/9783110863147
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