*(English)*Zbl 0877.34009

The paper treats the linear differential equation,

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 $(*)$ 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,

for $K>2k$ and an exceptional set, $E$, of finite linear measure. Herein $N(r,f)$ and $T(r,f)$ 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,

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 $max(m,n)$. Several theorems are proved regarding equation $(**)$, 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 |