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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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[13] doi:10.1515/9783110863147
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