This paper is concerned with the oscillatory behaviour of first-order delay differential equations of the form
\begin{equation}
x^{\prime}(t) + p(t)x(\tau(t)) = 0, \quad t \geqslant t_0,
\end{equation}
where $p, \tau \in C([t_0,\infty),{\rm I\!R}^+),{\rm I\!R}^+ = [0,\infty),\tau(t)$ is non-decreasing, $\tau(t) < t$ for $t \geqslant t_0$ and $\lim_{t\rightarrow\infty}\tau(t) = \infty$. Let the numbers $k$ and $L$ be defined by
\[
k = \mathop{{\lim\inf}}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds
\quad\hbox{and}\quad
L = \mathop{\lim\sup}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds.
\]
It is proved here that when $L < 1$ and $0 < k \leqslant 1/e$ all solutions of equation (1) oscillate in several cases in which the condition
\[
L > \frac{\ln\lambda_1 - 1 + \sqrt{5 - 2\lambda_1 + 2k\lambda_1}}{\lambda_1}
\]
holds, where $\lambda_1$ is the smaller root of the equation $\lambda = e^{k\lambda}$.
