For a convex function
f
∈
C
[
−
1
,
1
]
f \in C[ - 1,1]
we construct a sequence of convex polynomials
p
n
{p_n}
of degree not exceeding
n
n
such that
|
f
(
x
)
=
p
n
(
x
)
|
≤
C
ω
2
(
f
,
1
−
x
2
/
n
)
,
−
1
≤
x
≤
1
|f(x) = {p_n}(x)| \leq C{\omega _2}(f,\sqrt {1 - {x^2}} /n), - 1 \leq x \leq 1
. If in addition
f
f
is monotone it follows that the polynomials are also monotone thus providing simultaneous monotone and convex approximation.