Recently, we have extended the well-known Müntz-Szász theorem by showing that if
f
(
z
)
f(z)
is absolutely continuous and
|
f
′
(
x
)
|
⩾
k
>
0
|f’(x)| \geqslant k > 0
a.e. on
(
a
,
b
)
(a,b)
, where
a
⩾
0
a \geqslant 0
and if
{
n
p
}
\{ {n_p}\}
is a sequence of positive numbers tending to infinity and satisfying
∑
p
=
1
∞
1
/
n
p
=
∞
\sum _{p = 1}^\infty 1/{n_p} = \infty
, then the sequence
{
f
(
x
)
n
p
}
\{ f{(x)^{{n_p}}}\}
is complete on
(
a
,
b
)
(a,b)
if and only if
f
(
x
)
f(x)
is strictly monotone on
(
a
,
b
)
(a,b)
. We now apply Zarecki’s theorem to improve the condition "
|
f
′
(
x
)
|
⩾
k
>
0
|f’(x)| \geqslant k > 0
a.e. on
(
a
,
b
)
(a,b)
" by the condition
f
′
(
x
)
≠
0
f’(x) \ne 0
a.e. on
(
a
,
b
)
(a,b)
". Furthermore, we extend some well-known theorems of Picone, Mikusiński, and Boas.