On the hyperorder of solutions of higher order differential equations
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.
scientific article; zbMATH DE number 2052049