In this paper we study discretizations of the general pantograph equation
\[
y
′
(
t
)
=
a
y
(
t
)
+
b
y
(
θ
(
t
)
)
+
c
y
′
(
ϕ
(
t
)
)
,
t
≥
0
,
y
(
0
)
=
y
0
,
y’(t) = ay(t) + by(\theta (t)) + cy’(\phi (t)),\quad t \geq 0,\quad y(0) = {y_0},
\]
, where a, b, c, and
y
0
{y_0}
are complex numbers and where
θ
\theta
and
ϕ
\phi
are strictly increasing functions on the nonnegative reals with
θ
(
0
)
=
ϕ
(
0
)
=
0
\theta (0) = \phi (0) = 0
and
θ
(
t
)
>
t
,
ϕ
(
t
)
>
t
\theta (t) > t, \phi (t) > t
for positive t. Our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations, and we will identify conditions on a, b, c and the stepsize which imply that the solution sequence
{
y
n
}
n
=
0
∞
\{ {y_n}\} _{n=0}^\infty
is bounded or that it tends to zero algebraically, as a negative power of n.