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On the complex oscillation of some linear differential equations. (English) Zbl 0877.34009

The paper treats the linear differential equation,

${f}^{\left(k\right)}+A\left(z\right)f=0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $k\ge 2$ is an integer, $A\left(z\right)$ is a transcendental entire function of order $\sigma \left(A\right)$. It is shown that any non-trivial solution of $\left(*\right)$ satisfies $\lambda \left(f\right)\ge \sigma \left(A\right)$, where $\lambda \left(f\right)$ is the exponent of convergence of the zero-sequence of $f$, under the condition,

$K\overline{N}\left(r,\frac{1}{A}\right)\le T\left(r,A\right),\phantom{\rule{1.em}{0ex}}r\ne E$

for $K>2k$ and an exceptional set, $E$, of finite linear measure. Herein $N\left(r,f\right)$ and $T\left(r,f\right)$ denotes the counting function and the characteristic function of $f$ respectively. A very nice example is given demonstrating the result. Several technical lemmas are extremely well done preparing the proof of the theorem.

The other half of the paper treats the second-order equation,

${f}^{\text{'}\text{'}}+\left({e}^{{p}_{1}\left(z\right)}+{e}^{{p}_{2}\left(z\right)}+Q\left(z\right)\right)f=0,\phantom{\rule{2.em}{0ex}}\left(**\right)$

where ${p}_{1}\left(z\right)$ and ${p}_{2}\left(z\right)$ are non-constant polynomials of degree $n$ and $m$ respectively. $Q\left(z\right)$ is an entire function of order less than $max\left(m,n\right)$. Several theorems are proved regarding equation $\left(**\right)$, where once again several well done lemmas prepare the proof of the theorem. The paper is very well written.

##### MSC:
 34M10 Oscillation, growth of solutions (ODE in the complex domain) 30D05 Functional equations in the complex domain, iteration and composition of analytic functions