On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order. (English) Zbl 1141.34054

The authors study the growth of solutions of the linear differential equation \[ f^{(k)}+ A_{k-1}(z) f^{(k-1)}(z)+\cdots+ A_0f= 0,\tag{1} \] where \(A_0(z),\dots, A_{k-1}(z)\) are entire functions with \(A_0(z)\not\equiv 0\) \((k\geq 2)\). It is well-known that all solution of (1) are entire functions. Let \(\sigma(f)\) denote the order of growth of entire function \(f(z)\), \(\tau(f)\) to denote the type of \(f(z)\) with \(\sigma(f)=\sigma\) and use the notation \(\sigma_2(f)\) to denote the hyper-order of \(f(z)\).
Z.-X. Chen obtained the following result.
Proposition A. Let \(A_j(z)\) \((j= 0,\dots,k-1)\) be entire functions such that \[ \max\{\sigma(A_j), j=1,\dots, k-1\}< \sigma(A_0)<+ \infty. \] Then every solution \(f\not\equiv 0\) of (1) satisfies \(\sigma_2(f)= \sigma(A_0)\).
The authors proved the following theorem
Theorem 1. Let \(A_j(z)\) \((j= 0,\dots,k- 1)\) be entire functions satisfying \(\sigma(A_0)= \sigma\), \(\tau(A_0)=\tau\), \(0<\sigma<\infty\), \(0<\tau<\infty\), and let \(\sigma(A_j)\leq\sigma\), \(\tau(A_j)< \tau\) if \(\sigma(A_j)= \sigma\) \((j= 0,\dots, k-1)\), then every solution \(f\not\equiv 0\) of (1) satisfies \(\sigma_2(f)= \sigma(A_0)\).
The authors investigate in addition the case \(A_j(z)= h_j(z)\exp(P_j(z))\), where \(h_j(z)\) is an entire functions; \(P_j(z)\) \((j= 0,\dots, k- 1)\) are polynomials with degree \(n\geq 1\).


34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
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