## 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

Zbl 1131.34059
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