zbMATH — the first resource for mathematics

On the growth of solutions of certain linear differential equations. (English) Zbl 0759.34005
Suppose \(g_ j\), \(0\leq j\leq n-1\), and \(h\) are entire functions and that for some \(k\), \(0\leq k\leq n-1\), the order of \(g_ k\) does not exceed \({1\over 2}\) and does exceed the order of \(h\) and the order of all other \(g_ j\). It is shown that then every solution of the differential equation \(f^{(n)}+\sum_{j=0}^{n-1}g_ j f^{(j)}=h\) is either a polynomial or an entire function of infinite order. This generalizes a previous result of the author for second order equations.

34M99 Ordinary differential equations in the complex domain
30D20 Entire functions of one complex variable (general theory)
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Full Text: DOI