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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 $$n\geq 1$$.

##### MSC:
 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
##### Keywords:
entire function; hyper order
Full Text:
##### References:
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