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On growth k-order of solutions of a complex homogeneous linear differential equation. (English) Zbl 0652.34008
Let $$f=f(z)$$ be an entire function. The growth k-order $$\rho_ k=\rho_ k(f)$$ is defined as $$\rho_ k:=\limsup_{r\to \infty}(\log_{k+1}M(r)/\log r)$$, where $$M(r):=\max \{| f(z)| | | z| =r\}$$ and $$\log_{k+1}$$ is defined successively as $$\log_ 1r=\log r,...,\log_{k+1}r=\log (\log_ kr).$$ The growth index i(f) is defined as $$i(f)=0$$ if f(z) is a polynomial and $$i(f)=\min \{k\in {\mathbb{N}}| \rho_ k(f)<\infty \}$$ if f(z) is transcendental. Consider a linear differential equation $(1)\quad L(w):=w^{(n)}+a_ 1(z)w^{(n-1)}+...+a_ n(z)w=0$ where every $$a_ j$$ is entire and at least some $$a_ j$$ is not a constant. The author defines the k-grade of the singularity of (1) as $$\gamma_ k:=\sup \{\rho_ k(w)| L(w)=0\}$$ and the growth index of (1) as $$\delta:=\sup \{i(w)| L(w)=0\}$$. Let $$p:=\max \{i(a_ j)| j=1,...,n\}$$. In the case $$0<p<\infty$$, set $$\alpha:=\max \{\rho_ p(a_ j)| j=1,...,n\}$$. Then the following theorem is obtained. Theorem. (i) $$\delta \leq 1+p$$. (ii) If $$0<p<\infty$$, then $$\gamma_{p+1}\leq \alpha$$. (iii) If every $$a_ j$$ is a polynomial, then $$\gamma_ 1\leq 1+\max \{\deg a_ j| j=1,...,n\}.$$ (iv) Let $$n=2$$. Then, $$\delta =1+p$$. In addition, $$\gamma_{p+1}=\alpha$$ if $$p>0$$ and $$\gamma_ 1\geq (\deg a_ 1-1)/3$$ if $$p=0$$.
Reviewer: K.Takano

##### MSC:
 34M99 Ordinary differential equations in the complex domain 34A30 Linear ordinary differential equations and systems, general 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable