×

Growth of solutions of a class of complex differential equations. (English) Zbl 1173.34054

The author studies the growth of solutions of the linear differential equations \[ f^{(k)}-e^{P(z)}\cdot f=Q(z),\quad k\in\mathbb N.\tag{1} \]
G. G. Gundersen and L. Z. Yang obtained the following result:
Theorem 1. Let \(P\) be a nonconstant polynomial. Then every solution \(f\) of the differential equation
\[ f'-e^{P(z)}\cdot f=1\tag{2} \]
is an entire function of infinite order.
The hyperorder of growth of a meromorphic function \(f\) is defined by
\[ \sigma_2(f)=\limsup_{r\to\infty}\frac{\log^+\log^+T(r,f)}{\log r}. \]
The author proves the following theorems:
Theorem 2. Let \(P\) be a nonconstant entire function, let \(Q\) be a nonzero polynomial, and let \(f\) be any entire solution of the differential equation (1). If \(P\) is a polynomial, then \(f\) has infinite order and its hyperorder \(\sigma_2(f)\) is a positive integer not exceeding the degree of \(P\). If \(P\) is transcendental with order less than \(\frac12\), then the hyperorder of \(f\) is infinite.
Theorem 3. Let \(P(z)=\sum_{k=0}^\infty c_k z^{n_k}\) be a nonconstant entire function of finite lower order with Fabry gaps, that is, \(\frac{k}{n_k}\to 0,k\to\infty\), then every entire solution \(f\) of the differential equation (2) has infinite order and its hyperorder is a positive integer or infinity.

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

Citations:

Zbl 1131.34059
PDFBibTeX XMLCite
Full Text: DOI