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Some precise estimates of the hyper order of solutions of some complex linear differential equations. (English) Zbl 1140.34445

Summary: Let \(\rho(f)\) and \(\rho_2(f)\) denote respectively the order and the hyper order of an entire function \(f\). We obtain some precise estimates of the hyper order of solutions of the following higher order linear differential equations
\[ f^{(k)}+\sum^{k-1}_{j=0}A_j(z)e^{P_j(z)}f^{(j)}=0 \]
and
\[ f^{(k)}+\sum^{k-1}_{j=0}A_j(z)e^{P_j(z)}+B_j(z))f^{(j)}f^{(j)}=0 \]
where \(k\geq 2\), \(P_j(z)\) \((j=0,\dots,k-1)\) are nonconstant polynomials such that \(\deg P_j=n\) \((j=0.\dots,k-1)\) and \(A_j(z)(\not\equiv 0), B_j(z)(\not\equiv 0)\) \((j=0,\dots,k-1)\) are entire functions with \((j=0,\dots,k-1)\). Under some conditions, we prove that every solution \(f(z)\not\equiv 0\) of the above equations is of infinite order and \(\rho_2(f)=n\).

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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